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

Theorem mo2icl 3583
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 2817 . . . . . 6 (𝑦 = 𝐴 → (𝑥 = 𝑦𝑥 = 𝐴))
21imbi2d 331 . . . . 5 (𝑦 = 𝐴 → ((𝜑𝑥 = 𝑦) ↔ (𝜑𝑥 = 𝐴)))
32albidv 2011 . . . 4 (𝑦 = 𝐴 → (∀𝑥(𝜑𝑥 = 𝑦) ↔ ∀𝑥(𝜑𝑥 = 𝐴)))
43imbi1d 332 . . 3 (𝑦 = 𝐴 → ((∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑)))
5 19.8a 2217 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 mo2v 2639 . . . 4 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
75, 6sylibr 225 . . 3 (∀𝑥(𝜑𝑥 = 𝑦) → ∃*𝑥𝜑)
84, 7vtoclg 3459 . 2 (𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∃*𝑥𝜑))
9 eqvisset 3405 . . . . . 6 (𝑥 = 𝐴𝐴 ∈ V)
109imim2i 16 . . . . 5 ((𝜑𝑥 = 𝐴) → (𝜑𝐴 ∈ V))
1110con3rr3 152 . . . 4 𝐴 ∈ V → ((𝜑𝑥 = 𝐴) → ¬ 𝜑))
1211alimdv 2007 . . 3 𝐴 ∈ V → (∀𝑥(𝜑𝑥 = 𝐴) → ∀𝑥 ¬ 𝜑))
13 alnex 1861 . . . 4 (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
14 exmo 2657 . . . . 5 (∃𝑥𝜑 ∨ ∃*𝑥𝜑)
1514ori 879 . . . 4 (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
1613, 15sylbi 208 . . 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 1635   = wceq 1637  wex 1859  wcel 2156  ∃*wmo 2631  Vcvv 3391
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 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-12 2214  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-v 3393
This theorem is referenced by:  invdisj  4830  reusv1  5066  reusv2lem1  5067  opabiotafun  6480  fseqenlem2  9131  dfac2b  9236  dfac2OLD  9238  imasaddfnlem  16393  imasvscafn  16402  bnj149  31268
  Copyright terms: Public domain W3C validator