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Theorem copsex2g 5405
Description: Implicit substitution inference for ordered pairs. (Contributed by NM, 28-May-1995.) Use a similar proof to copsex4g 5407 to reduce axiom usage. (Revised by SN, 1-Sep-2024.)
Hypothesis
Ref Expression
copsex2g.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
copsex2g ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝑦,𝜓   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem copsex2g
StepHypRef Expression
1 eqcom 2745 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
2 vex 3433 . . . . . 6 𝑥 ∈ V
3 vex 3433 . . . . . 6 𝑦 ∈ V
42, 3opth 5389 . . . . 5 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴𝑦 = 𝐵))
51, 4bitri 274 . . . 4 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝐴𝑦 = 𝐵))
65anbi1i 624 . . 3 ((⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑))
762exbii 1851 . 2 (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑))
8 id 22 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝐴𝑦 = 𝐵))
9 copsex2g.1 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
108, 9cgsex2g 3472 . 2 ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜓))
117, 10bitrid 282 1 ((𝐴𝑉𝐵𝑊) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wex 1782  wcel 2106  cop 4567
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568
This theorem is referenced by:  opelopabga  5443  ov6g  7426  ltresr  10906
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