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Theorem 3optocl 5630
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 344 . . . 4 (⟨𝑣, 𝑢⟩ = 𝐶 → (((𝐴𝑅𝐵𝑅) → 𝜒) ↔ ((𝐴𝑅𝐵𝑅) → 𝜃)))
4 3optocl.2 . . . . . . 7 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
54imbi2d 344 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴 → (((𝑣𝐷𝑢𝐹) → 𝜑) ↔ ((𝑣𝐷𝑢𝐹) → 𝜓)))
6 3optocl.3 . . . . . . 7 (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓𝜒))
76imbi2d 344 . . . . . 6 (⟨𝑧, 𝑤⟩ = 𝐵 → (((𝑣𝐷𝑢𝐹) → 𝜓) ↔ ((𝑣𝐷𝑢𝐹) → 𝜒)))
8 3optocl.5 . . . . . . 7 (((𝑥𝐷𝑦𝐹) ∧ (𝑧𝐷𝑤𝐹) ∧ (𝑣𝐷𝑢𝐹)) → 𝜑)
983expia 1118 . . . . . 6 (((𝑥𝐷𝑦𝐹) ∧ (𝑧𝐷𝑤𝐹)) → ((𝑣𝐷𝑢𝐹) → 𝜑))
101, 5, 7, 92optocl 5629 . . . . 5 ((𝐴𝑅𝐵𝑅) → ((𝑣𝐷𝑢𝐹) → 𝜒))
1110com12 32 . . . 4 ((𝑣𝐷𝑢𝐹) → ((𝐴𝑅𝐵𝑅) → 𝜒))
121, 3, 11optocl 5628 . . 3 (𝐶𝑅 → ((𝐴𝑅𝐵𝑅) → 𝜃))
1312impcom 411 . 2 (((𝐴𝑅𝐵𝑅) ∧ 𝐶𝑅) → 𝜃)
14133impa 1107 1 ((𝐴𝑅𝐵𝑅𝐶𝑅) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  cop 4554   × cxp 5536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pr 5313
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-opab 5112  df-xp 5544
This theorem is referenced by:  ecopovtrn  8385
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