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Theorem bropabg 43313
Description: Equivalence for two classes related by an ordered-pair class abstraction. A generalization of brsslt 27845. (Contributed by RP, 26-Sep-2024.)
Hypotheses
Ref Expression
bropabg.xA (𝑥 = 𝐴 → (𝜑𝜓))
bropabg.yB (𝑦 = 𝐵 → (𝜓𝜒))
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
StepHypRef Expression
1 bropabg.R . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
21bropaex12 5780 . 2 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 bropabg.xA . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
4 bropabg.yB . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
53, 4, 1brabg 5549 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝜒))
62, 5biadanii 822 1 (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478   class class class wbr 5148  {copab 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695
This theorem is referenced by:  cantnfresb  43314  rp-brsslt  43413
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