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Theorem copsex2dv 32626
Description: Implicit substitution deduction for ordered pairs. (Contributed by Thierry Arnoux, 4-May-2025.)
Hypotheses
Ref Expression
copsex2dv.a (𝜑𝐴𝑈)
copsex2dv.b (𝜑𝐵𝑉)
copsex2dv.1 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
Assertion
Ref Expression
copsex2dv (𝜑 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem copsex2dv
StepHypRef Expression
1 copsex2dv.1 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝜓𝜒))
21ex 412 . . 3 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))
32alrimivv 1926 . 2 (𝜑 → ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))
4 copsex2dv.a . 2 (𝜑𝐴𝑈)
5 copsex2dv.b . 2 (𝜑𝐵𝑉)
6 copsex2t 5503 . 2 ((∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)) ∧ (𝐴𝑈𝐵𝑉)) → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
73, 4, 5, 6syl12anc 837 1 (𝜑 → (∃𝑥𝑦(⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜓) ↔ 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  cop 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638
This theorem is referenced by:  brab2d  32627
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