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Theorem copsex2ga 5654
 Description: Implicit substitution inference for ordered pairs. Compare copsex2g 5356. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
copsex2ga.1 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
Assertion
Ref Expression
copsex2ga (𝐴 ∈ (𝑉 × 𝑊) → (𝜑 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem copsex2ga
StepHypRef Expression
1 xpss 5544 . . 3 (𝑉 × 𝑊) ⊆ (V × V)
21sseli 3890 . 2 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3 copsex2ga.1 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
43copsex2gb 5653 . . 3 (∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ (𝐴 ∈ (V × V) ∧ 𝜑))
54baibr 540 . 2 (𝐴 ∈ (V × V) → (𝜑 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
62, 5syl 17 1 (𝐴 ∈ (𝑉 × 𝑊) → (𝜑 ↔ ∃𝑥𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3409  ⟨cop 4531   × cxp 5526 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5099  df-xp 5534 This theorem is referenced by: (None)
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