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Theorem mosubopt 5172
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 2196 . . 3 𝑦𝑦𝑧∃*𝑥𝜑
2 nfe1 2195 . . . 4 𝑦𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
32nfmo 2656 . . 3 𝑦∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
4 nfa1 2196 . . . . 5 𝑧𝑧∃*𝑥𝜑
5 nfe1 2195 . . . . . . 7 𝑧𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
65nfex 2332 . . . . . 6 𝑧𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
76nfmo 2656 . . . . 5 𝑧∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
8 copsexg 5152 . . . . . . . 8 (𝐴 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ ∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
98mobidv 2644 . . . . . . 7 (𝐴 = ⟨𝑦, 𝑧⟩ → (∃*𝑥𝜑 ↔ ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
109biimpcd 240 . . . . . 6 (∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
1110sps 2221 . . . . 5 (∀𝑧∃*𝑥𝜑 → (𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
124, 7, 11exlimd 2255 . . . 4 (∀𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
1312sps 2221 . . 3 (∀𝑦𝑧∃*𝑥𝜑 → (∃𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
141, 3, 13exlimd 2255 . 2 (∀𝑦𝑧∃*𝑥𝜑 → (∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
15 simpl 470 . . . . . 6 ((𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → 𝐴 = ⟨𝑦, 𝑧⟩)
16152eximi 1920 . . . . 5 (∃𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
1716exlimiv 2021 . . . 4 (∃𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) → ∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩)
1817con3i 151 . . 3 (¬ ∃𝑦𝑧 𝐴 = ⟨𝑦, 𝑧⟩ → ¬ ∃𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
19 exmo 2664 . . . 4 (∃𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑) ∨ ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
2019ori 879 . . 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 172 1 (∀𝑦𝑧∃*𝑥𝜑 → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wal 1635   = wceq 1637  wex 1859  ∃*wmo 2633  cop 4383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384
This theorem is referenced by:  mosubop  5173  funoprabg  6992
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