Theorem exmoeu 1455

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 1425	. . . 4 |- (E*xph <-> (E.xph -> E!xph))
2	1	biimpi 158	. . 3 |- (E*xph -> (E.xph -> E!xph))
3	2	com12 11	. 2 |- (E.xph -> (E*xph -> E!xph))
4	1	biimpri 159	. . . 4 |- ((E.xph -> E!xph) -> E*xph)
5		euex 1436	. . . 4 |- (E!xph -> E.xph)
6	4, 5	imim12i 18	. . 3 |- ((E*xph -> E!xph) -> ((E.xph -> E!xph) -> E.xph))
7		peirce 85	. . 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 164	1 |- (E.xph <-> (E*xph -> E!xph))

Syntax hints:   -> wi 3   <-> wb 153  E.wex 1021  E!weu 1422  E*wmo 1423

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425

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