Theorem exmoeu 2246

Description: Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.)

Assertion

Ref	Expression
exmoeu	|- (E.xph <-> (E*xph -> E!xph))

Proof of Theorem exmoeu

Step	Hyp	Ref	Expression
1		df-mo 2209	. . . 4 |- (E*xph <-> (E.xph -> E!xph))
2	1	biimpi 186	. . 3 |- (E*xph -> (E.xph -> E!xph))
3	2	com12 27	. 2 |- (E.xph -> (E*xph -> E!xph))
4	1	biimpri 197	. . . 4 |- ((E.xph -> E!xph) -> E*xph)
5		euex 2227	. . . 4 |- (E!xph -> E.xph)
6	4, 5	imim12i 53	. . 3 |- ((E*xph -> E!xph) -> ((E.xph -> E!xph) -> E.xph))
7		peirce 172	. . 3 |- (((E.xph -> E!xph) -> E.xph) -> E.xph)
8	6, 7	syl 15	. 2 |- ((E*xph -> E!xph) -> E.xph)
9	3, 8	impbii 180	1 |- (E.xph <-> (E*xph -> E!xph))

Syntax hints:   -> wi 4   <-> wb 176  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: (None)