Theorem riotabiia 7397

Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 3416 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 2760	. 2 ⊢ V = V
2		riotabiia.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))
3	2	adantl 487	. . 3 ⊢ ((V = V ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓))
4	3	riotabidva 7396	. 2 ⊢ (V = V → (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓))
5	1, 4	ax-mp 5	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑥 ∈ 𝐴 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ℩crio 7376

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-uni 4867  df-iota 6494  df-riota 7377

This theorem is used by:  riotaxfrd  7411  lubfval  18527  glbfval  18540  odulub  18584  oduglb  18586  cnlnadjlem5  32678  cdj3lem3  33045  cdj3lem3b  33047  lshpkrlem1  40260  cdleme25cv  41508  cdlemk35  42062