Theorem 2euex 2366

Description: Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
2euex

Proof of Theorem 2euex

Step	Hyp	Ref	Expression
1		eu5 2310	. 2
2		excom 1849	. . . 4
3		nfe1 1840	. . . . . 6
4	3	nfmo 2301	. . . . 5
5		19.8a 1857	. . . . . . 7
6	5	moimi 2340	. . . . . 6
7		df-mo 2287	. . . . . 6
8	6, 7	sylib 196	. . . . 5
9	4, 8	eximd 1882	. . . 4
10	2, 9	syl5bi 217	. . 3
11	10	impcom 430	. 2
12	1, 11	sylbi 195	1

Syntax hints:  ->wi 4  /\wa 369  E.wex 1612  E!weu 2282  E*wmo 2283

This theorem is referenced by:  2exeu  2371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-eu 2286  df-mo 2287