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Theorem mo2icl 3352
Description: Theorem for inferring "at most one." (Contributed by NM, 17-Oct-1996.)
Assertion
Ref Expression
mo2icl (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem mo2icl
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2621 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
21imbi2d 329 . . . . 5 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
32albidv 1836 . . . 4 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
43imbi1d 330 . . 3 (𝑦 = 𝐴 → ((∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)))
5 19.8a 2039 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 mo2v 2465 . . . 4 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
75, 6sylibr 223 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
84, 7vtoclg 3239 . 2 (𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑))
9 eqvisset 3184 . . . . . 6 (𝑥 = 𝐴𝐴 ∈ V)
109imim2i 16 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
1110con3rr3 150 . . . 4 𝐴 ∈ V → ((𝜑𝑥 = 𝐴) → ¬ 𝜑))
1211alimdv 1832 . . 3 𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∀𝑥 ¬ 𝜑))
13 alnex 1697 . . . 4 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
14 exmo 2483 . . . . 5 (∃𝑥𝜑 ∨ ∃*𝑥𝜑)
1514ori 389 . . . 4 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
1613, 15sylbi 206 . . 3 (∀𝑥 ¬ 𝜑 → ∃*𝑥𝜑)
1712, 16syl6 34 . 2 𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑))
188, 17pm2.61i 175 1 (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1473   = wceq 1475  wex 1695  wcel 1977  ∃*wmo 2459  Vcvv 3173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175
This theorem is referenced by:  invdisj  4571  reusv1  4792  reusv2lem1  4794  opabiotafun  6169  fseqenlem2  8731  dfac2  8836  imasaddfnlem  16011  imasvscafn  16020  bnj149  30199
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