Theorem 2euex

Description: Double quantification with existential uniqueness. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker 2euexv when possible. (Contributed by NM, 3-Dec-2001) (Proof shortened by Andrew Salmon, 9-Jul-2011) (New usage is discouraged.)

	Ref	Expression
Assertion	2euex	|- ( E! x E. y ph -> E. y E! x ph )

Proof

Step	Hyp	Ref	Expression
1		df-eu	|- ( E! x E. y ph <-> ( E. x E. y ph /\ E* x E. y ph ) )
2		excom	|- ( E. x E. y ph <-> E. y E. x ph )
3		nfe1	|- F/ y E. y ph
4	3	nfmo	|- F/ y E* x E. y ph
5		19.8a	|- ( ph -> E. y ph )
6	5	moimi	|- ( E* x E. y ph -> E* x ph )
7		moeu	|- ( E* x ph <-> ( E. x ph -> E! x ph ) )
8	6 7	sylib	|- ( E* x E. y ph -> ( E. x ph -> E! x ph ) )
9	4 8	eximd	|- ( E* x E. y ph -> ( E. y E. x ph -> E. y E! x ph ) )
10	2 9	biimtrid	|- ( E* x E. y ph -> ( E. x E. y ph -> E. y E! x ph ) )
11	10	impcom	|- ( ( E. x E. y ph /\ E* x E. y ph ) -> E. y E! x ph )
12	1 11	sylbi	|- ( E! x E. y ph -> E. y E! x ph )