Theorem amosym1 32742

Description: A symmetry with ∃*.

See negsym1 32733 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		df-mo 2635	. 2 ⊢ (∃*𝑥∃*𝑥⊥ ↔ (∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥))
2		mof 32726	. . . . 5 ⊢ ∃*𝑥⊥
3		19.8a 2217	. . . . . 6 ⊢ (∃*𝑥⊥ → ∃𝑥∃*𝑥⊥)
4	3	notnotd 140	. . . . 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 139	. . . . . 6 ⊢ ¬ ¬ ∃*𝑥⊥
8	7	nex 1882	. . . . 5 ⊢ ¬ ∃𝑥 ¬ ∃*𝑥⊥
9		eunex 5059	. . . . 5 ⊢ (∃!𝑥∃*𝑥⊥ → ∃𝑥 ¬ ∃*𝑥⊥)
10	8, 9	mto 188	. . . 4 ⊢ ¬ ∃!𝑥∃*𝑥⊥
11	10	pm2.21i 117	. . 3 ⊢ (∃!𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
12	6, 11	ja 174	. 2 ⊢ ((∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥) → ∃*𝑥𝜑)
13	1, 12	sylbi 208	1 ⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ⊥wfal 1650  ∃wex 1859  ∃!weu 2630  ∃*wmo 2631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-nul 4983  ax-pow 5035

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-eu 2634  df-mo 2635

This theorem is referenced by: (None)