Theorem unqsym1 31594

Description: A symmetry with ∃!.

See negsym1 31586 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 31576	. . . 4 ⊢ ¬ ∃!𝑥⊥
2	1	nex 1722	. . 3 ⊢ ¬ ∃𝑥∃!𝑥⊥
3		euex 2482	. . 3 ⊢ (∃!𝑥∃!𝑥⊥ → ∃𝑥∃!𝑥⊥)
4	2, 3	mto 187	. 2 ⊢ ¬ ∃!𝑥∃!𝑥⊥
5	4	pm2.21i 115	1 ⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)

Syntax hints:   → wi 4  ⊥wfal 1480  ∃wex 1695  ∃!weu 2458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875

This theorem depends on definitions:  df-bi 196  df-tru 1478  df-fal 1481  df-ex 1696  df-eu 2462

This theorem is referenced by: (None)