Theorem 2euex 2276

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

Assertion

Ref	Expression
2euex	|- (E!xE.yph -> E.yE!xph)

Proof of Theorem 2euex

Step	Hyp	Ref	Expression
1		eu5 2242	. 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
2		excom 1741	. . . 4 |- (E.xE.yph <-> E.yE.xph)
3		nfe1 1732	. . . . . 6 |- F/yE.yph
4	3	nfmo 2221	. . . . 5 |- F/yE*xE.yph
5		19.8a 1756	. . . . . . 7 |- (ph -> E.yph)
6	5	moimi 2251	. . . . . 6 |- (E*xE.yph -> E*xph)
7		df-mo 2209	. . . . . 6 |- (E*xph <-> (E.xph -> E!xph))
8	6, 7	sylib 188	. . . . 5 |- (E*xE.yph -> (E.xph -> E!xph))
9	4, 8	eximd 1770	. . . 4 |- (E*xE.yph -> (E.yE.xph -> E.yE!xph))
10	2, 9	syl5bi 208	. . 3 |- (E*xE.yph -> (E.xE.yph -> E.yE!xph))
11	10	impcom 419	. 2 |- ((E.xE.yph /\ E*xE.yph) -> E.yE!xph)
12	1, 11	sylbi 187	1 |- (E!xE.yph -> E.yE!xph)

Syntax hints:   -> wi 4   /\ wa 358  E.wex 1541  E!weu 2204  E*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  2exeu  2281