Theorem mo2 1393

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 1376	. 2 |- (E*xph <-> (E.xph -> E!xph))
2		alnex 1029	. . . . 5 |- (A.x -. ph <-> -. E.xph)
3		pm2.21 76	. . . . . . 7 |- (-. ph -> (ph -> x = y))
4	3	19.20i 989	. . . . . 6 |- (A.x -. ph -> A.x(ph -> x = y))
5		19.8a 1025	. . . . . 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 201	. . . 4 |- (-. E.xph -> E.yA.x(ph -> x = y))
8		mo2.1	. . . . 5 |- (ph -> A.yph)
9	8	eumo0 1388	. . . 4 |- (E!xph -> E.yA.x(ph -> x = y))
10	7, 9	ja 137	. . 3 |- ((E.xph -> E!xph) -> E.yA.x(ph -> x = y))
11	8	eu3 1390	. . . . 5 |- (E!xph <-> (E.xph /\ E.yA.x(ph -> x = y)))
12	11	biimpr 152	. . . 4 |- ((E.xph /\ E.yA.x(ph -> x = y)) -> E!xph)
13	12	expcom 374	. . 3 |- (E.yA.x(ph -> x = y) -> (E.xph -> E!xph))
14	10, 13	impbi 157	. 2 |- ((E.xph -> E!xph) <-> E.yA.x(ph -> x = y))
15	1, 14	bitr 173	1 |- (E*xph <-> E.yA.x(ph -> x = y))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953  E.wex 977  E!weu 1373  E*wmo 1374

This theorem is referenced by:  mo3 1394  eu5 1402  immo 1410  moimv 1412  moanim 1420  mo2icl 1914  moabex 2756  dffun3 3513  dffunmof 3516  grothprim 8722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376

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