Theorem exmoeu 1809

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 1776	. . . 4 |- (E*xph <-> (E.xph -> E!xph))
2	1	biimpi 168	. . 3 |- (E*xph -> (E.xph -> E!xph))
3	2	com12 14	. 2 |- (E.xph -> (E*xph -> E!xph))
4	1	biimpri 169	. . . 4 |- ((E.xph -> E!xph) -> E*xph)
5		euex 1788	. . . 4 |- (E!xph -> E.xph)
6	4, 5	imim12i 21	. . 3 |- ((E*xph -> E!xph) -> ((E.xph -> E!xph) -> E.xph))
7		peirce 98	. . 3 |- (((E.xph -> E!xph) -> E.xph) -> E.xph)
8	6, 7	syl 12	. 2 |- ((E*xph -> E!xph) -> E.xph)
9	3, 8	impbii 174	1 |- (E.xph <-> (E*xph -> E!xph))

Syntax hints:   -> wi 3   <-> wb 163  E.wex 1326  E!weu 1771  E*wmo 1772

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776

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