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Theorem mo2icl 3535
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 2781 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
21imbi2d 329 . . . . 5 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
32albidv 2000 . . . 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 2205 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 mo2v 2624 . . . 4 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
75, 6sylibr 224 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
84, 7vtoclg 3415 . 2 (𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑))
9 eqvisset 3360 . . . . . 6 (𝑥 = 𝐴𝐴 ∈ V)
109imim2i 16 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
1110con3rr3 152 . . . 4 𝐴 ∈ V → ((𝜑𝑥 = 𝐴) → ¬ 𝜑))
1211alimdv 1996 . . 3 𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∀𝑥 ¬ 𝜑))
13 alnex 1853 . . . 4 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
14 exmo 2642 . . . . 5 (∃𝑥𝜑 ∨ ∃*𝑥𝜑)
1514ori 841 . . . 4 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
1613, 15sylbi 207 . . 3 (∀𝑥 ¬ 𝜑 → ∃*𝑥𝜑)
1712, 16syl6 35 . 2 𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑))
188, 17pm2.61i 176 1 (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1628   = wceq 1630  wex 1851  wcel 2144  ∃*wmo 2618  Vcvv 3349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-v 3351
This theorem is referenced by:  invdisj  4770  reusv1  4994  reusv2lem1  4996  opabiotafun  6401  fseqenlem2  9047  dfac2b  9152  dfac2OLD  9154  imasaddfnlem  16395  imasvscafn  16404  bnj149  31277
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