Theorem bropabg 43768

Description: Equivalence for two classes related by an ordered-pair class abstraction. A generalization of brslts 27772. (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 5709	. 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 5481	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜒))
6	2, 5	biadanii 827	1 ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431   class class class wbr 5072  {copab 5134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624

This theorem is referenced by:  cantnfresb  43769  rp-brsslt  43867