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

Theorem mosubopt 5529
Description: "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)
Assertion
Ref Expression
mosubopt (∀𝑦𝑧∃*𝑥𝜑 → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
Distinct variable group:   𝑥,𝑦,𝑧,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem mosubopt
StepHypRef Expression
1 nfa1 2152 . . 3 𝑦𝑦𝑧∃*𝑥𝜑
2 nfe1 2151 . . . 4 𝑦𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
32nfmov 2563 . . 3 𝑦∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
4 nfa1 2152 . . . . 5 𝑧𝑧∃*𝑥𝜑
5 nfe1 2151 . . . . . . 7 𝑧𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
65nfex 2328 . . . . . 6 𝑧𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
76nfmov 2563 . . . . 5 𝑧∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
8 copsexgw 5510 . . . . . . . 8 (𝐴 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
98mobidv 2552 . . . . . . 7 (𝐴 = ⟨𝑦, 𝑧⟩ → (∃*𝑥𝜑 ↔ ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
109biimpcd 249 . . . . . 6 (∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
1110sps 2186 . . . . 5 (∀𝑧∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
124, 7, 11exlimd 2219 . . . 4 (∀𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
1312sps 2186 . . 3 (∀𝑦𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
141, 3, 13exlimd 2219 . 2 (∀𝑦𝑧∃*𝑥𝜑 → (∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
15 simpl 482 . . . . 5 ((𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → 𝐴 = ⟨𝑦, 𝑧⟩)
16152eximi 1834 . . . 4 (∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
1716exlimiv 1929 . . 3 (∃𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
18 nexmo 2544 . . 3 (¬ ∃𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
1917, 18nsyl5 159 . 2 (¬ ∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
2014, 19pm2.61d1 180 1 (∀𝑦𝑧∃*𝑥𝜑 → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wex 1777  ∃*wmo 2541  cop 4654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655
This theorem is referenced by:  mosubop  5530  funoprabg  7571
  Copyright terms: Public domain W3C validator