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Theorem optocl 5731
Description: Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)
Hypotheses
Ref Expression
optocl.1 𝐷 = (𝐵 × 𝐶)
optocl.2 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
optocl.3 ((𝑥𝐵𝑦𝐶) → 𝜑)
Assertion
Ref Expression
optocl (𝐴𝐷𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem optocl
StepHypRef Expression
1 elxp3 5703 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
2 opelxp 5674 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) ↔ (𝑥𝐵𝑦𝐶))
3 optocl.3 . . . . . . 7 ((𝑥𝐵𝑦𝐶) → 𝜑)
42, 3sylbi 216 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜑)
5 optocl.2 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
64, 5imbitrid 243 . . . . 5 (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜓))
76imp 408 . . . 4 ((⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
87exlimivv 1936 . . 3 (∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
91, 8sylbi 216 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝜓)
10 optocl.1 . 2 𝐷 = (𝐵 × 𝐶)
119, 10eleq2s 2856 1 (𝐴𝐷𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  cop 4597   × cxp 5636
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5173  df-xp 5644
This theorem is referenced by:  2optocl  5732  3optocl  5733  ecoptocl  8753  ax1rid  11104  axcnre  11107
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