Theorem 2euex 2133

Description: Double quantification with existential uniqueness. (The proof was 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 2099	. 2 |- (E!xE.yph <-> (E.xE.yph /\ E*xE.yph))
2		excom 1711	. . . 4 |- (E.xE.yph <-> E.yE.xph)
3		hbe1 1681	. . . . . 6 |- (E.yph -> A.yE.yph)
4	3	hbmo 2097	. . . . 5 |- (E*xE.yph -> A.yE*xE.yph)
5		19.8a 1694	. . . . . . 7 |- (ph -> E.yph)
6	5	immoi 2108	. . . . . 6 |- (E*xE.yph -> E*xph)
7		df-mo 2071	. . . . . 6 |- (E*xph <-> (E.xph -> E!xph))
8	6, 7	sylib 263	. . . . 5 |- (E*xE.yph -> (E.xph -> E!xph))
9	4, 8	eximd 1727	. . . 4 |- (E*xE.yph -> (E.yE.xph -> E.yE!xph))
10	2, 9	syl5bi 270	. . 3 |- (E*xE.yph -> (E.xE.yph -> E.yE!xph))
11	10	impcom 490	. 2 |- ((E.xE.yph /\ E*xE.yph) -> E.yE!xph)
12	1, 11	sylbi 237	1 |- (E!xE.yph -> E.yE!xph)

Syntax hints:   -> wi 3   /\ wa 433  E.wex 1644  E!weu 2066  E*wmo 2067

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1621  ax-gen 1622  ax-8 1623  ax-9 1624  ax-10 1625  ax-11 1626  ax-12 1627  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-10o 1810  ax-16 1883  ax-11o 1893

This theorem depends on definitions:  df-bi 232  df-or 434  df-an 435  df-ex 1645  df-sb 1845  df-eu 2070  df-mo 2071

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