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Theorem optocl 5772
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 5744 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
2 opelxp 5714 . . . . . . 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 406 . . . 4 ((⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
87exlimivv 1928 . . 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 2847 1 (𝐴𝐷𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wex 1774  wcel 2099  cop 4635   × cxp 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5211  df-xp 5684
This theorem is referenced by:  2optocl  5773  3optocl  5774  ecoptocl  8826  ax1rid  11185  axcnre  11188
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