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Theorem opelopab2a 4155
Description: Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypothesis
Ref Expression
opelopabga.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
opelopab2a ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opelopab2a
StepHypRef Expression
1 eleq1 2178 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐶𝐴𝐶))
2 eleq1 2178 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐷𝐵𝐷))
31, 2bi2anan9 578 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → ((𝑥𝐶𝑦𝐷) ↔ (𝐴𝐶𝐵𝐷)))
4 opelopabga.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
53, 4anbi12d 462 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (((𝑥𝐶𝑦𝐷) ∧ 𝜑) ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓)))
65opelopabga 4153 . 2 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓)))
76bianabs 583 1 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  cop 3498  {copab 3956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958
This theorem is referenced by:  opelopab2  4160  brab2a  4560  brab2ga  4582  ltdfpr  7278
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