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Theorem copsex2dv 5499
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 1928 . 2 (𝜑 → ∀𝑥𝑦((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))
4 copsex2dv.a . 2 (𝜑𝐴𝑈)
5 copsex2dv.b . 2 (𝜑𝐵𝑉)
6 copsex2t 5497 . 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 1538   = wceq 1540  wex 1779  wcel 2108  cop 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633
This theorem is referenced by:  brab2d  32619  brab2dd  48741
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