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Theorem mosubopt 4532
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 1486 . . 3 𝑦𝑦𝑧∃*𝑥𝜑
2 nfe1 1437 . . . 4 𝑦𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
32nfmo 1975 . . 3 𝑦∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
4 nfa1 1486 . . . . 5 𝑧𝑧∃*𝑥𝜑
5 nfe1 1437 . . . . . . 7 𝑧𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
65nfex 1580 . . . . . 6 𝑧𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
76nfmo 1975 . . . . 5 𝑧∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
8 copsexg 4095 . . . . . . . 8 (𝐴 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
98mobidv 1991 . . . . . . 7 (𝐴 = ⟨𝑦, 𝑧⟩ → (∃*𝑥𝜑 ↔ ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
109biimpcd 158 . . . . . 6 (∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
1110sps 1482 . . . . 5 (∀𝑧∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
124, 7, 11exlimd 1540 . . . 4 (∀𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
1312sps 1482 . . 3 (∀𝑦𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
141, 3, 13exlimd 1540 . 2 (∀𝑦𝑧∃*𝑥𝜑 → (∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
15 moanimv 2030 . . 3 (∃*𝑥(∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)) ↔ (∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
16 simpl 108 . . . . . 6 ((𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → 𝐴 = ⟨𝑦, 𝑧⟩)
17162eximi 1544 . . . . 5 (∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
1817ancri 318 . . . 4 (∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → (∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
1918moimi 2020 . . 3 (∃*𝑥(∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ ∧ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)) → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
2015, 19sylbir 134 . 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 103  wal 1294   = wceq 1296  wex 1433  ∃*wmo 1956  cop 3469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475
This theorem is referenced by:  mosubop  4533  funoprabg  5782
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