Users' Mathboxes Mathbox for Anthony Hart < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  unqsym1 Structured version   Visualization version   GIF version

Theorem unqsym1 32863
Description: A symmetry with ∃!.

See negsym1 32855 for more information. (Contributed by Anthony Hart, 6-Sep-2011.)

Assertion
Ref Expression
unqsym1 (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)

Proof of Theorem unqsym1
StepHypRef Expression
1 neufal 32845 . . . 4 ¬ ∃!𝑥
21nex 1895 . . 3 ¬ ∃𝑥∃!𝑥
3 euex 2591 . . 3 (∃!𝑥∃!𝑥⊥ → ∃𝑥∃!𝑥⊥)
42, 3mto 188 . 2 ¬ ∃!𝑥∃!𝑥
54pm2.21i 117 1 (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1665  wex 1874  ∃!weu 2581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890
This theorem depends on definitions:  df-bi 198  df-an 385  df-tru 1656  df-fal 1666  df-ex 1875  df-eu 2582
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator