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Theorem bropabg 41687
Description: Equivalence for two classes related by an ordered-pair class abstraction. A generalization of brsslt 27147. (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 5728 . 2 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 bropabg.xA . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
4 bropabg.yB . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
53, 4, 1brabg 5501 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵𝜒))
62, 5biadanii 821 1 (𝐴𝑅𝐵 ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3448   class class class wbr 5110  {copab 5172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644
This theorem is referenced by:  cantnfresb  41688  rp-brsslt  41769
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