Theorem unqsym1 32549

Description: A symmetry with ∃!.

See negsym1 32541 for more information. (Contributed by Anthony Hart, 6-Sep-2011.)

Assertion

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

Proof of Theorem unqsym1

Step	Hyp	Ref	Expression
1		unnf 32531	. . . 4 ⊢ ¬ ∃!𝑥⊥
2	1	nex 1771	. . 3 ⊢ ¬ ∃𝑥∃!𝑥⊥
3		euex 2522	. . 3 ⊢ (∃!𝑥∃!𝑥⊥ → ∃𝑥∃!𝑥⊥)
4	2, 3	mto 188	. 2 ⊢ ¬ ∃!𝑥∃!𝑥⊥
5	4	pm2.21i 116	1 ⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)

Syntax hints:   → wi 4  ⊥wfal 1528  ∃wex 1744  ∃!weu 2498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945

This theorem depends on definitions:  df-bi 197  df-tru 1526  df-fal 1529  df-ex 1745  df-eu 2502

This theorem is referenced by: (None)