Theorem exmoeu 1039

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 1010	. . . 4 |- (E*xph <-> (E.xph -> E!xph))
2	1	biimp 133	. . 3 |- (E*xph -> (E.xph -> E!xph))
3	2	com12 13	. 2 |- (E.xph -> (E*xph -> E!xph))
4	1	biimpr 134	. . . 4 |- ((E.xph -> E!xph) -> E*xph)
5		euex 1021	. . . 4 |- (E!xph -> E.xph)
6	4, 5	syl34 20	. . 3 |- ((E*xph -> E!xph) -> ((E.xph -> E!xph) -> E.xph))
7		peirce 76	. . 3 |- (((E.xph -> E!xph) -> E.xph) -> E.xph)
8	6, 7	syl 12	. 2 |- ((E*xph -> E!xph) -> E.xph)
9	3, 8	impbi 139	1 |- (E.xph <-> (E*xph -> E!xph))

Syntax hints:   -> wi 2   <-> wb 127  E.wex 678  E!weu 1007  E*wmo 1008

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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