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Theorem 3optocl 5120
 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 329 . . . 4 (⟨𝑣, 𝑢⟩ = 𝐶 → (((𝐴𝑅𝐵𝑅) → 𝜒) ↔ ((𝐴𝑅𝐵𝑅) → 𝜃)))
4 3optocl.2 . . . . . . 7 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
54imbi2d 329 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴 → (((𝑣𝐷𝑢𝐹) → 𝜑) ↔ ((𝑣𝐷𝑢𝐹) → 𝜓)))
6 3optocl.3 . . . . . . 7 (⟨𝑧, 𝑤⟩ = 𝐵 → (𝜓𝜒))
76imbi2d 329 . . . . . 6 (⟨𝑧, 𝑤⟩ = 𝐵 → (((𝑣𝐷𝑢𝐹) → 𝜓) ↔ ((𝑣𝐷𝑢𝐹) → 𝜒)))
8 3optocl.5 . . . . . . 7 (((𝑥𝐷𝑦𝐹) ∧ (𝑧𝐷𝑤𝐹) ∧ (𝑣𝐷𝑢𝐹)) → 𝜑)
983expia 1259 . . . . . 6 (((𝑥𝐷𝑦𝐹) ∧ (𝑧𝐷𝑤𝐹)) → ((𝑣𝐷𝑢𝐹) → 𝜑))
101, 5, 7, 92optocl 5119 . . . . 5 ((𝐴𝑅𝐵𝑅) → ((𝑣𝐷𝑢𝐹) → 𝜒))
1110com12 32 . . . 4 ((𝑣𝐷𝑢𝐹) → ((𝐴𝑅𝐵𝑅) → 𝜒))
121, 3, 11optocl 5118 . . 3 (𝐶𝑅 → ((𝐴𝑅𝐵𝑅) → 𝜃))
1312impcom 445 . 2 (((𝐴𝑅𝐵𝑅) ∧ 𝐶𝑅) → 𝜃)
14133impa 1251 1 ((𝐴𝑅𝐵𝑅𝐶𝑅) → 𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   × cxp 5036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044 This theorem is referenced by:  ecopovtrn  7737
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