Theorem riotabidva 7388

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3438 analog.) (Contributed by NM, 17-Jan-2012.)

Hypothesis

Ref	Expression
riotabidva.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))

Assertion

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

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem riotabidva

Step	Hyp	Ref	Expression
1		riotabidva.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
2	1	pm5.32da 578	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
3	2	iotabidv 6528	. 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
4		df-riota 7368	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
5		df-riota 7368	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))
6	3, 4, 5	3eqtr4g 2796	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ℩cio 6494  ℩crio 7367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3956  df-ss 3966  df-uni 4910  df-iota 6496  df-riota 7368

This theorem is referenced by:  riotabiia  7389  dfceil2  13809  cidpropd  17659  grpinvpropd  18935  mirval  28170  mirfv  28171  grpoidval  30030  adjval2  31408  riotaeqbidva  32000  xdivval  32349  toslub  32407  tosglb  32409  ringinvval  32651  glbconN  38551  glbconNOLD  38552  cdlemk33N  40084  cdlemk34  40085  cdlemkid4  40109