Theorem riotabiia 7408

Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3437 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 2735	. 2 ⊢ V = V
2		riotabiia.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))
3	2	adantl 481	. . 3 ⊢ ((V = V ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓))
4	3	riotabidva 7407	. 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 206   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ℩crio 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-uni 4913  df-iota 6516  df-riota 7388

This theorem is referenced by:  riotaxfrd  7422  lubfval  18408  glbfval  18421  odulub  18465  oduglb  18467  cnlnadjlem5  32100  cdj3lem3  32467  cdj3lem3b  32469  lshpkrlem1  39092  cdleme25cv  40341  cdlemk35  40895