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Theorem optocl 5104
Description: Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.)
Hypotheses
Ref Expression
optocl.1 𝐷 = (𝐵 × 𝐶)
optocl.2 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
optocl.3 ((𝑥𝐵𝑦𝐶) → 𝜑)
Assertion
Ref Expression
optocl (𝐴𝐷𝜓)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem optocl
StepHypRef Expression
1 elxp3 5078 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
2 opelxp 5056 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) ↔ (𝑥𝐵𝑦𝐶))
3 optocl.3 . . . . . . 7 ((𝑥𝐵𝑦𝐶) → 𝜑)
42, 3sylbi 205 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜑)
5 optocl.2 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
64, 5syl5ib 232 . . . . 5 (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜓))
76imp 443 . . . 4 ((⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
87exlimivv 1845 . . 3 (∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
91, 8sylbi 205 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝜓)
10 optocl.1 . 2 𝐷 = (𝐵 × 𝐶)
119, 10eleq2s 2701 1 (𝐴𝐷𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1975  cop 4126   × cxp 5022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-opab 4634  df-xp 5030
This theorem is referenced by:  2optocl  5105  3optocl  5106  ecoptocl  7697  ax1rid  9834  axcnre  9837
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