MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  optocl Structured version   Visualization version   GIF version

Theorem optocl 5779
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 5750 . . 3 (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))
2 opelxp 5720 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) ↔ (𝑥𝐵𝑦𝐶))
3 optocl.3 . . . . . . 7 ((𝑥𝐵𝑦𝐶) → 𝜑)
42, 3sylbi 217 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜑)
5 optocl.2 . . . . . 6 (⟨𝑥, 𝑦⟩ = 𝐴 → (𝜑𝜓))
64, 5imbitrid 244 . . . . 5 (⟨𝑥, 𝑦⟩ = 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) → 𝜓))
76imp 406 . . . 4 ((⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
87exlimivv 1931 . . 3 (∃𝑥𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) → 𝜓)
91, 8sylbi 217 . 2 (𝐴 ∈ (𝐵 × 𝐶) → 𝜓)
10 optocl.1 . 2 𝐷 = (𝐵 × 𝐶)
119, 10eleq2s 2858 1 (𝐴𝐷𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wex 1778  wcel 2107  cop 4631   × cxp 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205  df-xp 5690
This theorem is referenced by:  2optocl  5780  3optocl  5781  ecoptocl  8848  ax1rid  11202  axcnre  11205
  Copyright terms: Public domain W3C validator