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Theorem 2optocl 5764
Description: Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
Hypotheses
Ref Expression
2optocl.1 𝑅 = (𝐶 × 𝐷)
2optocl.2 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
2optocl.3 (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓𝜒))
2optocl.4 (((𝑥𝐶𝑦𝐷) ∧ (𝑧𝐶𝑤𝐷)) → 𝜑)
Assertion
Ref Expression
2optocl ((𝐴𝑅𝐵𝑅) → 𝜒)
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝐴   𝑧,𝐵,𝑤   𝑥,𝐶,𝑦,𝑧,𝑤   𝑥,𝐷,𝑦,𝑧,𝑤   𝜓,𝑥,𝑦   𝜒,𝑧,𝑤   𝑧,𝑅,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑧,𝑤)   𝜒(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem 2optocl
StepHypRef Expression
1 2optocl.1 . . 3 𝑅 = (𝐶 × 𝐷)
2 2optocl.3 . . . 4 (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓𝜒))
32imbi2d 340 . . 3 (⟨𝑧, 𝑤⟩ = 𝐵 → ((𝐴𝑅𝜓) ↔ (𝐴𝑅𝜒)))
4 2optocl.2 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
54imbi2d 340 . . . . 5 (⟨𝑥, 𝑦⟩ = 𝐴 → (((𝑧𝐶𝑤𝐷) → 𝜑) ↔ ((𝑧𝐶𝑤𝐷) → 𝜓)))
6 2optocl.4 . . . . . 6 (((𝑥𝐶𝑦𝐷) ∧ (𝑧𝐶𝑤𝐷)) → 𝜑)
76ex 412 . . . . 5 ((𝑥𝐶𝑦𝐷) → ((𝑧𝐶𝑤𝐷) → 𝜑))
81, 5, 7optocl 5763 . . . 4 (𝐴𝑅 → ((𝑧𝐶𝑤𝐷) → 𝜓))
98com12 32 . . 3 ((𝑧𝐶𝑤𝐷) → (𝐴𝑅𝜓))
101, 3, 9optocl 5763 . 2 (𝐵𝑅 → (𝐴𝑅𝜒))
1110impcom 407 1 ((𝐴𝑅𝐵𝑅) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  cop 4629   × cxp 5667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204  df-xp 5675
This theorem is referenced by:  3optocl  5765  ecopovsym  8812  axaddf  11139  axmulf  11140
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