Theorem riotaeqbidv 5649

Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)

Hypotheses

Ref	Expression
riotaeqbidv.1	⊢ (𝜑 → 𝐴 = 𝐵)
riotaeqbidv.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
riotaeqbidv	⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem riotaeqbidv

Step	Hyp	Ref	Expression
1		riotaeqbidv.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	riotabidv 5648	. 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))
3		riotaeqbidv.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
4	3	riotaeqdv 5647	. 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥 ∈ 𝐵 𝜒))
5	2, 4	eqtrd 2127	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1296  ℩crio 5645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-uni 3676  df-iota 5014  df-riota 5646

This theorem is referenced by:  acexmidlemab  5684