Theorem exmoeu 1411

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 1381	. . . 4 |- (E*xph <-> (E.xph -> E!xph))
2	1	biimp 151	. . 3 |- (E*xph -> (E.xph -> E!xph))
3	2	com12 11	. 2 |- (E.xph -> (E*xph -> E!xph))
4	1	biimpr 152	. . . 4 |- ((E.xph -> E!xph) -> E*xph)
5		euex 1392	. . . 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	impbi 157	1 |- (E.xph <-> (E*xph -> E!xph))

Syntax hints:   -> wi 3   <-> wb 146  E.wex 978  E!weu 1378  E*wmo 1379

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381

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