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Theorem 3optocl 5366
Description: Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.)
Hypotheses
Ref Expression
3optocl.1 𝑅 = (𝐷 × 𝐹)
3optocl.2 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
3optocl.3 (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓𝜒))
3optocl.4 (⟨𝑣, 𝑢⟩ = 𝐶 → (𝜒𝜃))
3optocl.5 (((𝑥𝐷𝑦𝐹) ∧ (𝑧𝐷𝑤𝐹) ∧ (𝑣𝐷𝑢𝐹)) → 𝜑)
Assertion
Ref Expression
3optocl ((𝐴𝑅𝐵𝑅𝐶𝑅) → 𝜃)
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝐴   𝑧,𝐵,𝑤,𝑣,𝑢   𝑣,𝐶,𝑢   𝑥,𝐷,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝐹,𝑦,𝑧,𝑤,𝑣,𝑢   𝑧,𝑅,𝑤,𝑣,𝑢   𝜓,𝑥,𝑦   𝜒,𝑧,𝑤   𝜃,𝑣,𝑢
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝜓(𝑧,𝑤,𝑣,𝑢)   𝜒(𝑥,𝑦,𝑣,𝑢)   𝜃(𝑥,𝑦,𝑧,𝑤)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦,𝑧,𝑤)   𝑅(𝑥,𝑦)

Proof of Theorem 3optocl
StepHypRef Expression
1 3optocl.1 . . . 4 𝑅 = (𝐷 × 𝐹)
2 3optocl.4 . . . . 5 (⟨𝑣, 𝑢⟩ = 𝐶 → (𝜒𝜃))
32imbi2d 331 . . . 4 (⟨𝑣, 𝑢⟩ = 𝐶 → (((𝐴𝑅𝐵𝑅) → 𝜒) ↔ ((𝐴𝑅𝐵𝑅) → 𝜃)))
4 3optocl.2 . . . . . . 7 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
54imbi2d 331 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴 → (((𝑣𝐷𝑢𝐹) → 𝜑) ↔ ((𝑣𝐷𝑢𝐹) → 𝜓)))
6 3optocl.3 . . . . . . 7 (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓𝜒))
76imbi2d 331 . . . . . 6 (⟨𝑧, 𝑤⟩ = 𝐵 → (((𝑣𝐷𝑢𝐹) → 𝜓) ↔ ((𝑣𝐷𝑢𝐹) → 𝜒)))
8 3optocl.5 . . . . . . 7 (((𝑥𝐷𝑦𝐹) ∧ (𝑧𝐷𝑤𝐹) ∧ (𝑣𝐷𝑢𝐹)) → 𝜑)
983expia 1150 . . . . . 6 (((𝑥𝐷𝑦𝐹) ∧ (𝑧𝐷𝑤𝐹)) → ((𝑣𝐷𝑢𝐹) → 𝜑))
101, 5, 7, 92optocl 5365 . . . . 5 ((𝐴𝑅𝐵𝑅) → ((𝑣𝐷𝑢𝐹) → 𝜒))
1110com12 32 . . . 4 ((𝑣𝐷𝑢𝐹) → ((𝐴𝑅𝐵𝑅) → 𝜒))
121, 3, 11optocl 5364 . . 3 (𝐶𝑅 → ((𝐴𝑅𝐵𝑅) → 𝜃))
1312impcom 396 . 2 (((𝐴𝑅𝐵𝑅) ∧ 𝐶𝑅) → 𝜃)
14133impa 1136 1 ((𝐴𝑅𝐵𝑅𝐶𝑅) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  cop 4339   × cxp 5274
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 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061
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 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-opab 4871  df-xp 5282
This theorem is referenced by:  ecopovtrn  8053
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