Theorem riotabiia 6021

Description: Equivalent wff's yield equal restricted iotas (inference form). (rabbiia 2798 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 2232	. 2 ⊢ V = V
2		riotabiia.1	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ 𝜓))
3	2	adantl 277	. . 3 ⊢ ((V = V ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝜓))
4	3	riotabidva 6020	. 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 105   = wceq 1398   ∈ wcel 2203  Vcvv 2812  ℩crio 6001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-uni 3914  df-iota 5311  df-riota 6002

This theorem is referenced by:  caucvgsrlemfv  8102  dfgcd3  12699