Theorem bropabg 43336

Description: Equivalence for two classes related by an ordered-pair class abstraction. A generalization of brsslt 27830. (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 5777	. 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 5544	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜒))
6	2, 5	biadanii 822	1 ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480   class class class wbr 5143  {copab 5205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691

This theorem is referenced by:  cantnfresb  43337  rp-brsslt  43436