Theorem riotabidva 7336

Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3406 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 579	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒)))
3	2	iotabidv 6477	. 2 ⊢ (𝜑 → (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)))
4		df-riota 7317	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
5		df-riota 7317	. 2 ⊢ (℩𝑥 ∈ 𝐴 𝜒) = (℩𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))
6	3, 4, 5	3eqtr4g 2797	1 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 𝜓) = (℩𝑥 ∈ 𝐴 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ℩cio 6447  ℩crio 7316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3443  df-ss 3919  df-uni 4865  df-iota 6449  df-riota 7317

This theorem is referenced by:  riotabiia  7337  dfceil2  13763  cidpropd  17637  grpinvpropd  18949  mirval  28731  mirfv  28732  grpoidval  30592  adjval2  31970  riotaeqbidva  32573  xdivval  33002  toslub  33057  tosglb  33059  ringinvval  33319  glbconN  39705  cdlemk33N  41237  cdlemk34  41238  cdlemkid4  41262