Theorem amosym1 33331

Description: A symmetry with ∃*.

See negsym1 33322 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 2603	. 2 ⊢ (∃*𝑥∃*𝑥⊥ ↔ (∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥))
2		mofal 33315	. . . . 5 ⊢ ∃*𝑥⊥
3		19.8a 2110	. . . . . 6 ⊢ (∃*𝑥⊥ → ∃𝑥∃*𝑥⊥)
4	3	notnotd 141	. . . . 5 ⊢ (∃*𝑥⊥ → ¬ ¬ ∃𝑥∃*𝑥⊥)
5	2, 4	ax-mp 5	. . . 4 ⊢ ¬ ¬ ∃𝑥∃*𝑥⊥
6	5	pm2.21i 117	. . 3 ⊢ (¬ ∃𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
7	2	notnoti 140	. . . . . 6 ⊢ ¬ ¬ ∃*𝑥⊥
8	7	nex 1764	. . . . 5 ⊢ ¬ ∃𝑥 ¬ ∃*𝑥⊥
9		eunex 5139	. . . . 5 ⊢ (∃!𝑥∃*𝑥⊥ → ∃𝑥 ¬ ∃*𝑥⊥)
10	8, 9	mto 189	. . . 4 ⊢ ¬ ∃!𝑥∃*𝑥⊥
11	10	pm2.21i 117	. . 3 ⊢ (∃!𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
12	6, 11	ja 175	. 2 ⊢ ((∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥) → ∃*𝑥𝜑)
13	1, 12	sylbi 209	1 ⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ⊥wfal 1520  ∃wex 1743  ∃*wmo 2546  ∃!weu 2584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-12 2107  ax-nul 5063  ax-pow 5115

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-fal 1521  df-ex 1744  df-nf 1748  df-mo 2548  df-eu 2585

This theorem is referenced by: (None)