Theorem bropabg 42630

Description: Equivalence for two classes related by an ordered-pair class abstraction. A generalization of brsslt 27669. (Contributed by RP, 26-Sep-2024.)

Hypotheses

Ref	Expression
bropabg.xA	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
bropabg.yB	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
bropabg.R	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Assertion

Ref	Expression
bropabg	⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem bropabg

Step	Hyp	Ref	Expression
1		bropabg.R	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2	1	bropaex12 5760	. 2 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3		bropabg.xA	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4		bropabg.yB	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
5	3, 4, 1	brabg 5532	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜒))
6	2, 5	biadanii 819	1 ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   class class class wbr 5141  {copab 5203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675

This theorem is referenced by:  cantnfresb  42631  rp-brsslt  42731