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Theorem 2optocl 5353
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 329 . . 3 (⟨𝑧, 𝑤⟩ = 𝐵 → ((𝐴𝑅𝜓) ↔ (𝐴𝑅𝜒)))
4 2optocl.2 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
54imbi2d 329 . . . . 5 (⟨𝑥, 𝑦⟩ = 𝐴 → (((𝑧𝐶𝑤𝐷) → 𝜑) ↔ ((𝑧𝐶𝑤𝐷) → 𝜓)))
6 2optocl.4 . . . . . 6 (((𝑥𝐶𝑦𝐷) ∧ (𝑧𝐶𝑤𝐷)) → 𝜑)
76ex 449 . . . . 5 ((𝑥𝐶𝑦𝐷) → ((𝑧𝐶𝑤𝐷) → 𝜑))
81, 5, 7optocl 5352 . . . 4 (𝐴𝑅 → ((𝑧𝐶𝑤𝐷) → 𝜓))
98com12 32 . . 3 ((𝑧𝐶𝑤𝐷) → (𝐴𝑅𝜓))
101, 3, 9optocl 5352 . 2 (𝐵𝑅 → (𝐴𝑅𝜒))
1110impcom 445 1 ((𝐴𝑅𝐵𝑅) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  cop 4327   × cxp 5264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272
This theorem is referenced by:  3optocl  5354  ecopovsym  8016  axaddf  10158  axmulf  10159
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