Theorem riotabiia 7391

Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3423 analog.) (Contributed by NM, 16-Jan-2012.)

Hypothesis

Ref	Expression
riotabiia.1	⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
riotabiia	⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓)

Proof of Theorem riotabiia

Step	Hyp	Ref	Expression
1		eqid 2725	. 2 ⊢ V = V
2		riotabiia.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))
3	2	adantl 480	. . 3 ⊢ ((V = V ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓))
4	3	riotabidva 7390	. 2 ⊢ (V = V → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓))
5	1, 4	ax-mp 5	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Vcvv 3463  ℩crio 7369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-in 3946  df-ss 3956  df-uni 4902  df-iota 6493  df-riota 7370

This theorem is referenced by:  riotaxfrd  7405  lubfval  18339  glbfval  18352  odulub  18396  oduglb  18398  cnlnadjlem5  31897  cdj3lem3  32264  cdj3lem3b  32266  lshpkrlem1  38610  cdleme25cv  39859  cdlemk35  40413

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