Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mo2icl Structured version   Visualization version   GIF version

Theorem mo2icl 3656
 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 2813 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
21imbi2d 344 . . . . 5 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
32albidv 1921 . . . 4 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
43imbi1d 345 . . 3 (𝑦 = 𝐴 → ((∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)))
5 19.8a 2179 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 df-mo 2601 . . . 4 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
75, 6sylibr 237 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
84, 7vtoclg 3518 . 2 (𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑))
9 eqvisset 3461 . . . . . 6 (𝑥 = 𝐴𝐴 ∈ V)
109imim2i 16 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
1110con3rr3 158 . . . 4 𝐴 ∈ V → ((𝜑𝑥 = 𝐴) → ¬ 𝜑))
1211alimdv 1917 . . 3 𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∀𝑥 ¬ 𝜑))
13 alnex 1783 . . . 4 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
14 nexmo 2602 . . . 4 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
1513, 14sylbi 220 . . 3 (∀𝑥 ¬ 𝜑 → ∃*𝑥𝜑)
1612, 15syl6 35 . 2 𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑))
178, 16pm2.61i 185 1 (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2112  ∃*wmo 2599  Vcvv 3444 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-mo 2601  df-clab 2780  df-cleq 2794  df-clel 2873  df-v 3446 This theorem is referenced by:  invdisj  5017  reusv1  5266  reusv2lem1  5267  opabiotafun  6723  fseqenlem2  9440  dfac2b  9545  imasaddfnlem  16796  imasvscafn  16805  bnj149  32255
 Copyright terms: Public domain W3C validator