Theorem amosym1 34615

Description: A symmetry with ∃*.

See negsym1 34606 for more information. (Contributed by Anthony Hart, 13-Sep-2011.)

Assertion

Ref	Expression
amosym1	⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)

Proof of Theorem amosym1

Step	Hyp	Ref	Expression
1		moeu 2583	. 2 ⊢ (∃*𝑥∃*𝑥⊥ ↔ (∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥))
2		mofal 34598	. . . . 5 ⊢ ∃*𝑥⊥
3		19.8a 2174	. . . . . 6 ⊢ (∃*𝑥⊥ → ∃𝑥∃*𝑥⊥)
4	3	notnotd 144	. . . . 5 ⊢ (∃*𝑥⊥ → ¬ ¬ ∃𝑥∃*𝑥⊥)
5	2, 4	ax-mp 5	. . . 4 ⊢ ¬ ¬ ∃𝑥∃*𝑥⊥
6	5	pm2.21i 119	. . 3 ⊢ (¬ ∃𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
7	2	notnoti 143	. . . . . 6 ⊢ ¬ ¬ ∃*𝑥⊥
8	7	nex 1803	. . . . 5 ⊢ ¬ ∃𝑥 ¬ ∃*𝑥⊥
9		eunex 5313	. . . . 5 ⊢ (∃!𝑥∃*𝑥⊥ → ∃𝑥 ¬ ∃*𝑥⊥)
10	8, 9	mto 196	. . . 4 ⊢ ¬ ∃!𝑥∃*𝑥⊥
11	10	pm2.21i 119	. . 3 ⊢ (∃!𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
12	6, 11	ja 186	. 2 ⊢ ((∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥) → ∃*𝑥𝜑)
13	1, 12	sylbi 216	1 ⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ⊥wfal 1551  ∃wex 1782  ∃*wmo 2538  ∃!weu 2568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-nul 5230  ax-pow 5288

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-mo 2540  df-eu 2569

This theorem is referenced by: (None)