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Theorem unqsym1 33798
Description: A symmetry with ∃!.

See negsym1 33790 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 33779 . . . 4 ¬ ∃!𝑥
21nex 1802 . . 3 ¬ ∃𝑥∃!𝑥
3 euex 2663 . . 3 (∃!𝑥∃!𝑥⊥ → ∃𝑥∃!𝑥⊥)
42, 3mto 200 . 2 ¬ ∃!𝑥∃!𝑥
54pm2.21i 119 1 (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1550  wex 1781  ∃!weu 2654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-fal 1551  df-ex 1782  df-eu 2655
This theorem is referenced by: (None)
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