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Theorem optocl 4615
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 4593 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
2 opelxp 4569 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) ↔ (𝑥𝐵𝑦𝐶))
3 optocl.3 . . . . . . 7 ((𝑥𝐵𝑦𝐶) → 𝜑)
42, 3sylbi 120 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜑)
5 optocl.2 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
64, 5syl5ib 153 . . . . 5 (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜓))
76imp 123 . . . 4 ((⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
87exlimivv 1868 . . 3 (∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
91, 8sylbi 120 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝜓)
10 optocl.1 . 2 𝐷 = (𝐵 × 𝐶)
119, 10eleq2s 2234 1 (𝐴𝐷𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480  cop 3530   × cxp 4537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545
This theorem is referenced by:  2optocl  4616  3optocl  4617  ecoptocl  6516  ax1rid  7692  ax0id  7693  axcnre  7696
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