cbvixpv - Metamath Proof Explorer

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Theorem cbvixpv 8933

Description: Change bound variable in an indexed Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.)

**Hypothesis**
Ref	Expression
cbvixpv.1	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

**Assertion**
Ref	Expression
cbvixpv	⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Distinct variable groups: 𝑥,𝐴,𝑦 𝑦,𝐵 𝑥,𝐶 Allowed substitution hints: 𝐵(𝑥) 𝐶(𝑦)

**Proof of Theorem cbvixpv**
Step	Hyp	Ref	Expression
1		nfcv 2899	. 2 ⊢ Ⅎ𝑦𝐵
2		nfcv 2899	. 2 ⊢ Ⅎ𝑥𝐶
3		cbvixpv.1	. 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
4	1, 2, 3	cbvixp 8932	1 ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 Xcixp 8915

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

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fn 6551 df-fv 6556 df-ixp 8916

This theorem is referenced by: funcpropd 17888 invfuc 17965 natpropd 17967 dprdw 19966 dprdwd 19967 ptuni2 23479 ptbasin 23480 ptbasfi 23484 ptpjopn 23515 ptclsg 23518 dfac14 23521 ptcnp 23525 ptcmplem2 23956 ptcmpg 23960 prdsxmslem2 24437 upixp 37202 rrxsnicc 45688 ioorrnopn 45693 ioorrnopnxr 45695 ovnsubadd 45960 hoidmvlelem4 45986 hoidmvle 45988 hspdifhsp 46004 hoiqssbllem2 46011 hspmbl 46017 hoimbl 46019 opnvonmbl 46022 ovnovollem3 46046

	
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