Theorem mo2 1439

Description: Alternate definition of "at most one."

Hypothesis

Ref	Expression
mo2.1	|- (ph -> A.yph)

Assertion

Ref	Expression
mo2	|- (E*xph <-> E.yA.x(ph -> x = y))

Distinct variable group:   x,y

Proof of Theorem mo2

Step	Hyp	Ref	Expression
1		df-mo 1422	. 2 |- (E*xph <-> (E.xph -> E!xph))
2		alnex 1069	. . . . 5 |- (A.x -. ph <-> -. E.xph)
3		pm2.21 76	. . . . . . 7 |- (-. ph -> (ph -> x = y))
4	3	19.20i 1028	. . . . . 6 |- (A.x -. ph -> A.x(ph -> x = y))
5		19.8a 1065	. . . . . 6 |- (A.x(ph -> x = y) -> E.yA.x(ph -> x = y))
6	4, 5	syl 10	. . . . 5 |- (A.x -. ph -> E.yA.x(ph -> x = y))
7	2, 6	sylbir 199	. . . 4 |- (-. E.xph -> E.yA.x(ph -> x = y))
8		mo2.1	. . . . 5 |- (ph -> A.yph)
9	8	eumo0 1434	. . . 4 |- (E!xph -> E.yA.x(ph -> x = y))
10	7, 9	ja 135	. . 3 |- ((E.xph -> E!xph) -> E.yA.x(ph -> x = y))
11	8	eu3 1436	. . . . 5 |- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))
12	11	biimpri 150	. . . 4 |- ((E.xph /\ E.yA.x(ph -> x = y)) -> E!xph)
13	12	expcom 372	. . 3 |- (E.yA.x(ph -> x = y) -> (E.xph -> E!xph))
14	10, 13	impbii 155	. 2 |- ((E.xph -> E!xph) <-> E.yA.x(ph -> x = y))
15	1, 14	bitri 171	1 |- (E*xph <-> E.yA.x(ph -> x = y))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992  E.wex 1016  E!weu 1419  E*wmo 1420

This theorem is referenced by:  mo3 1440  eu5 1448  immo 1456  moimv 1458  moanim 1466  mo2icl 1969  moabex 2844  dffun3 3632  dffun6f 3635  grothprim 9057  sbmo 11787

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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