Theorem amosym1 33776

Description: A symmetry with ∃*.

See negsym1 33767 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 2668	. 2 ⊢ (∃*𝑥∃*𝑥⊥ ↔ (∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥))
2		mofal 33759	. . . . 5 ⊢ ∃*𝑥⊥
3		19.8a 2180	. . . . . 6 ⊢ (∃*𝑥⊥ → ∃𝑥∃*𝑥⊥)
4	3	notnotd 146	. . . . 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 145	. . . . . 6 ⊢ ¬ ¬ ∃*𝑥⊥
8	7	nex 1801	. . . . 5 ⊢ ¬ ∃𝑥 ¬ ∃*𝑥⊥
9		eunex 5293	. . . . 5 ⊢ (∃!𝑥∃*𝑥⊥ → ∃𝑥 ¬ ∃*𝑥⊥)
10	8, 9	mto 199	. . . 4 ⊢ ¬ ∃!𝑥∃*𝑥⊥
11	10	pm2.21i 119	. . 3 ⊢ (∃!𝑥∃*𝑥⊥ → ∃*𝑥𝜑)
12	6, 11	ja 188	. 2 ⊢ ((∃𝑥∃*𝑥⊥ → ∃!𝑥∃*𝑥⊥) → ∃*𝑥𝜑)
13	1, 12	sylbi 219	1 ⊢ (∃*𝑥∃*𝑥⊥ → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ⊥wfal 1549  ∃wex 1780  ∃*wmo 2620  ∃!weu 2653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2177  ax-nul 5212  ax-pow 5268

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-mo 2622  df-eu 2654

This theorem is referenced by: (None)