itgeq2dv - Metamath Proof Explorer

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Theorem itgeq2dv 25710

Description: Equality theorem for an integral. (Contributed by Mario Carneiro, 7-Jul-2014.)

**Hypothesis**
Ref	Expression
itgeq2dv.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)

**Assertion**
Ref	Expression
itgeq2dv	⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)

Distinct variable group: 𝜑,𝑥 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥) 𝐶(𝑥)

**Proof of Theorem itgeq2dv**
Step	Hyp	Ref	Expression
1		itgeq2dv.1	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶)
2	1	ralrimiva 3143	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)
3		itgeq2 25706	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)
4	2, 3	syl 17	1 ⊢ (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ∫citg 25546

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-seq 13999 df-sum 15665 df-itg 25551

This theorem is referenced by: itgmpt 25711 itgneg 25732 itgss2 25741 itgconst 25747 itgaddlem2 25752 itgadd 25753 itgsub 25754 itgfsum 25755 itgmulc2lem2 25761 itgmulc2 25762 itgabs 25763 ftc1lem4 25973 ftc2ditglem 25979 itgparts 25981 itgsubstlem 25982 itgsubst 25983 itgpowd 25984 itgulm 26343 itgulm2 26344 areaval 26895 circlemeth 34272 circlemethnat 34273 circlevma 34274 circlemethhgt 34275 hgt749d 34281 itgaddnclem2 37152 itgaddnc 37153 itgsubnc 37155 itgmulc2nclem2 37160 itgmulc2nc 37161 itgabsnc 37162 ftc1cnnclem 37164 areacirc 37186 3factsumint2 41493 3factsumint4 41495 lcmineqlem1 41500 lcmineqlem3 41502 lcmineqlem10 41509 lcmineqlem12 41511 lcmineqlem13 41512 intlewftc 41532 areaquad 42644 itgsin0pilem1 45338 itgsinexplem1 45342 itgsinexp 45343 ditgeqiooicc 45348 ditgeq3d 45352 itgcoscmulx 45357 itgsincmulx 45362 itgioocnicc 45365 itgiccshift 45368 itgperiod 45369 wallispilem1 45453 wallispilem2 45454 dirkeritg 45490 fourierdlem16 45511 fourierdlem21 45516 fourierdlem30 45525 fourierdlem73 45567 fourierdlem81 45575 fourierdlem82 45576 fourierdlem83 45577 fourierdlem87 45581 fourierdlem93 45587 fourierdlem95 45589 fourierdlem101 45595 fourierdlem103 45597 fourierdlem104 45598 fourierdlem111 45605 fourierdlem112 45606 fourierdlem115 45609 sqwvfoura 45616 sqwvfourb 45617 etransclem46 45668

	
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