Theorem exmoeu 1452

Description: Existence in terms of "at most one" and uniqueness.

Assertion

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

Proof of Theorem exmoeu

Step	Hyp	Ref	Expression
1		df-mo 1422	. . . 4 |- (E*xph <-> (E.xph -> E!xph))
2	1	biimpi 149	. . 3 |- (E*xph -> (E.xph -> E!xph))
3	2	com12 11	. 2 |- (E.xph -> (E*xph -> E!xph))
4	1	biimpri 150	. . . 4 |- ((E.xph -> E!xph) -> E*xph)
5		euex 1433	. . . 4 |- (E!xph -> E.xph)
6	4, 5	imim12i 18	. . 3 |- ((E*xph -> E!xph) -> ((E.xph -> E!xph) -> E.xph))
7		peirce 82	. . 3 |- (((E.xph -> E!xph) -> E.xph) -> E.xph)
8	6, 7	syl 10	. 2 |- ((E*xph -> E!xph) -> E.xph)
9	3, 8	impbii 155	1 |- (E.xph <-> (E*xph -> E!xph))

Syntax hints:   -> wi 3   <-> wb 144  E.wex 1016  E!weu 1419  E*wmo 1420

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422

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