Theorem unqsym1 37340

Description: A symmetry with ∃!.

See negsym1 37332 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		neufal 37321	. . . 4 ⊢ ¬ ∃!𝑥⊥
2	1	nex 1832	. . 3 ⊢ ¬ ∃𝑥∃!𝑥⊥
3		euex 2605	. . 3 ⊢ (∃!𝑥∃!𝑥⊥ → ∃𝑥∃!𝑥⊥)
4	2, 3	mto 200	. 2 ⊢ ¬ ∃!𝑥∃!𝑥⊥
5	4	pm2.21i 120	1 ⊢ (∃!𝑥∃!𝑥⊥ → ∃!𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ⊥wfal 1581  ∃wex 1811  ∃!weu 2596

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-eu 2597

This theorem is used by: (None)