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Theorem copsex2gb 5799
Description: Implicit substitution inference for ordered pairs. Compare copsex2ga 5800. (Contributed by NM, 12-Mar-2014.)
Hypothesis
Ref Expression
copsex2ga.1 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
Assertion
Ref Expression
copsex2gb (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)

Proof of Theorem copsex2gb
StepHypRef Expression
1 elvv 5743 . . 3 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
21anbi1i 623 . 2 ((𝐴 ∈ (V × V) ∧ 𝜑) ↔ (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
3 19.41vv 1946 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
4 copsex2ga.1 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
54pm5.32i 574 . . 3 ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
652exbii 1843 . 2 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
72, 3, 63bitr2ri 300 1 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  Vcvv 3468  cop 4629   × cxp 5667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204  df-xp 5675
This theorem is referenced by:  copsex2ga  5800  elopaba  5801  dfxrn2  37759  elcnvlem  42928
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