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Theorem mosubopt 5001
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 2068 . . 3 𝑦𝑦𝑧∃*𝑥𝜑
2 nfe1 2067 . . . 4 𝑦𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
32nfmo 2515 . . 3 𝑦∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
4 nfa1 2068 . . . . 5 𝑧𝑧∃*𝑥𝜑
5 nfe1 2067 . . . . . . 7 𝑧𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
65nfex 2192 . . . . . 6 𝑧𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
76nfmo 2515 . . . . 5 𝑧∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
8 copsexg 4985 . . . . . . . 8 (𝐴 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
98mobidv 2519 . . . . . . 7 (𝐴 = ⟨𝑦, 𝑧⟩ → (∃*𝑥𝜑 ↔ ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
109biimpcd 239 . . . . . 6 (∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
1110sps 2093 . . . . 5 (∀𝑧∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
124, 7, 11exlimd 2125 . . . 4 (∀𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
1312sps 2093 . . 3 (∀𝑦𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
141, 3, 13exlimd 2125 . 2 (∀𝑦𝑧∃*𝑥𝜑 → (∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
15 simpl 472 . . . . . 6 ((𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → 𝐴 = ⟨𝑦, 𝑧⟩)
16152eximi 1803 . . . . 5 (∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
1716exlimiv 1898 . . . 4 (∃𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
1817con3i 150 . . 3 (¬ ∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ¬ ∃𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
19 exmo 2523 . . . 4 (∃𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) ∨ ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
2019ori 389 . . 3 (¬ ∃𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
2118, 20syl 17 . 2 (¬ ∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
2214, 21pm2.61d1 171 1 (∀𝑦𝑧∃*𝑥𝜑 → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521   = wceq 1523  wex 1744  ∃*wmo 2499  cop 4216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217
This theorem is referenced by:  mosubop  5002  funoprabg  6801
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