Theorem amosym1

Description: A symmetry with E* .

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

	Ref	Expression
Assertion	amosym1	$$⊢{\exists }^{*}x{\exists }^{*}x\perp \to {\exists }^{*}x\phi$$

Proof

Step	Hyp	Ref	Expression
1		moeu	$$⊢{\exists }^{*}x{\exists }^{*}x\perp \leftrightarrow \left(\exists x{\exists }^{*}x\perp \to \exists !x{\exists }^{*}x\perp \right)$$
2		mofal	$$⊢{\exists }^{*}x\perp$$
3		19.8a	$$⊢{\exists }^{*}x\perp \to \exists x{\exists }^{*}x\perp$$
4	3	notnotd	$$⊢{\exists }^{*}x\perp \to \neg \neg \exists x{\exists }^{*}x\perp$$
5	2 4	ax-mp	$$⊢\neg \neg \exists x{\exists }^{*}x\perp$$
6	5	pm2.21i	$$⊢\neg \exists x{\exists }^{*}x\perp \to {\exists }^{*}x\phi$$
7	2	notnoti	$$⊢\neg \neg {\exists }^{*}x\perp$$
8	7	nex	$$⊢\neg \exists x\neg {\exists }^{*}x\perp$$
9		eunex	$$⊢\exists !x{\exists }^{*}x\perp \to \exists x\neg {\exists }^{*}x\perp$$
10	8 9	mto	$$⊢\neg \exists !x{\exists }^{*}x\perp$$
11	10	pm2.21i	$$⊢\exists !x{\exists }^{*}x\perp \to {\exists }^{*}x\phi$$
12	6 11	ja	$$⊢\left(\exists x{\exists }^{*}x\perp \to \exists !x{\exists }^{*}x\perp \right)\to {\exists }^{*}x\phi$$
13	1 12	sylbi	$$⊢{\exists }^{*}x{\exists }^{*}x\perp \to {\exists }^{*}x\phi$$