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Theorem 2optocl 4699
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 230 . . 3 (⟨𝑧, 𝑤⟩ = 𝐵 → ((𝐴𝑅𝜓) ↔ (𝐴𝑅𝜒)))
4 2optocl.2 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
54imbi2d 230 . . . . 5 (⟨𝑥, 𝑦⟩ = 𝐴 → (((𝑧𝐶𝑤𝐷) → 𝜑) ↔ ((𝑧𝐶𝑤𝐷) → 𝜓)))
6 2optocl.4 . . . . . 6 (((𝑥𝐶𝑦𝐷) ∧ (𝑧𝐶𝑤𝐷)) → 𝜑)
76ex 115 . . . . 5 ((𝑥𝐶𝑦𝐷) → ((𝑧𝐶𝑤𝐷) → 𝜑))
81, 5, 7optocl 4698 . . . 4 (𝐴𝑅 → ((𝑧𝐶𝑤𝐷) → 𝜓))
98com12 30 . . 3 ((𝑧𝐶𝑤𝐷) → (𝐴𝑅𝜓))
101, 3, 9optocl 4698 . 2 (𝐵𝑅 → (𝐴𝑅𝜒))
1110impcom 125 1 ((𝐴𝑅𝐵𝑅) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  cop 3594   × cxp 4620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-opab 4062  df-xp 4628
This theorem is referenced by:  3optocl  4700  ecopovsym  6624  ecopovsymg  6627  th3qlem2  6631  axaddcom  7847
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