Theorem mo2 1026

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(s):   x,y

Proof of Theorem mo2

Step	Hyp	Ref	Expression
1		df-mo 1010	. 2 |- (E*xph <-> (E.xph -> E!xph))
2		alnex 716	. . . . 5 |- (A.x -. ph <-> -. E.xph)
3		pm2.21 71	. . . . . . 7 |- (-. ph -> (ph -> x = y))
4	3	19.20i 691	. . . . . 6 |- (A.x -. ph -> A.x(ph -> x = y))
5		19.8a 712	. . . . . 6 |- (A.x(ph -> x = y) -> E.yA.x(ph -> x = y))
6	4, 5	syl 12	. . . . 5 |- (A.x -. ph -> E.yA.x(ph -> x = y))
7	2, 6	sylbir 176	. . . 4 |- (-. E.xph -> E.yA.x(ph -> x = y))
8		mo2.1	. . . . 5 |- (ph -> A.yph)
9	8	eumo0 1022	. . . 4 |- (E!xph -> E.yA.x(ph -> x = y))
10	7, 9	ja 118	. . 3 |- ((E.xph -> E!xph) -> E.yA.x(ph -> x = y))
11	8	eu3 1024	. . . . . 6 |- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))
12	11	biimpr 134	. . . . 5 |- ((E.xph /\ E.yA.x(ph -> x = y)) -> E!xph)
13	12	exp 291	. . . 4 |- (E.xph -> (E.yA.x(ph -> x = y) -> E!xph))
14	13	com12 13	. . 3 |- (E.yA.x(ph -> x = y) -> (E.xph -> E!xph))
15	10, 14	impbi 139	. 2 |- ((E.xph -> E!xph) <-> E.yA.x(ph -> x = y))
16	1, 15	bitr 151	1 |- (E*xph <-> E.yA.x(ph -> x = y))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678   = weq 797  E!weu 1007  E*wmo 1008

This theorem is referenced by:  mo3 1027  eu5 1035  immo 1043  moimv 1044  moanim 1051  mo2icl 1434  moabex 1868  dffun3 2675  dffunmof 2678

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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