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Theorem 2optocl 5366
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 331 . . 3 (⟨𝑧, 𝑤⟩ = 𝐵 → ((𝐴𝑅𝜓) ↔ (𝐴𝑅𝜒)))
4 2optocl.2 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
54imbi2d 331 . . . . 5 (⟨𝑥, 𝑦⟩ = 𝐴 → (((𝑧𝐶𝑤𝐷) → 𝜑) ↔ ((𝑧𝐶𝑤𝐷) → 𝜓)))
6 2optocl.4 . . . . . 6 (((𝑥𝐶𝑦𝐷) ∧ (𝑧𝐶𝑤𝐷)) → 𝜑)
76ex 401 . . . . 5 ((𝑥𝐶𝑦𝐷) → ((𝑧𝐶𝑤𝐷) → 𝜑))
81, 5, 7optocl 5365 . . . 4 (𝐴𝑅 → ((𝑧𝐶𝑤𝐷) → 𝜓))
98com12 32 . . 3 ((𝑧𝐶𝑤𝐷) → (𝐴𝑅𝜓))
101, 3, 9optocl 5365 . 2 (𝐵𝑅 → (𝐴𝑅𝜒))
1110impcom 396 1 ((𝐴𝑅𝐵𝑅) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  cop 4340   × cxp 5275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-opab 4872  df-xp 5283
This theorem is referenced by:  3optocl  5367  ecopovsym  8053  axaddf  10219  axmulf  10220
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