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Theorem mosubopt 4461
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 1475 . . 3 𝑦𝑦𝑧∃*𝑥𝜑
2 nfe1 1426 . . . 4 𝑦𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
32nfmo 1963 . . 3 𝑦∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
4 nfa1 1475 . . . . 5 𝑧𝑧∃*𝑥𝜑
5 nfe1 1426 . . . . . . 7 𝑧𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
65nfex 1569 . . . . . 6 𝑧𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
76nfmo 1963 . . . . 5 𝑧∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
8 copsexg 4035 . . . . . . . 8 (𝐴 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
98mobidv 1979 . . . . . . 7 (𝐴 = ⟨𝑦, 𝑧⟩ → (∃*𝑥𝜑 ↔ ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
109biimpcd 157 . . . . . 6 (∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
1110sps 1471 . . . . 5 (∀𝑧∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
124, 7, 11exlimd 1529 . . . 4 (∀𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
1312sps 1471 . . 3 (∀𝑦𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
141, 3, 13exlimd 1529 . 2 (∀𝑦𝑧∃*𝑥𝜑 → (∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
15 moanimv 2018 . . 3 (∃*𝑥(∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)) ↔ (∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
16 simpl 107 . . . . . 6 ((𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → 𝐴 = ⟨𝑦, 𝑧⟩)
17162eximi 1533 . . . . 5 (∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
1817ancri 317 . . . 4 (∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → (∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
1918moimi 2008 . . 3 (∃*𝑥(∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)) → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
2015, 19sylbir 133 . 2 ((∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)) → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
2114, 20syl 14 1 (∀𝑦𝑧∃*𝑥𝜑 → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wal 1283   = wceq 1285  wex 1422  ∃*wmo 1944  cop 3425
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431
This theorem is referenced by:  mosubop  4462  funoprabg  5679
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