Theorem mo2 1377

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 1360	. 2 |- (E*xph <-> (E.xph -> E!xph))
2		alnex 1009	. . . . 5 |- (A.x -. ph <-> -. E.xph)
3		pm2.21 76	. . . . . . 7 |- (-. ph -> (ph -> x = y))
4	3	19.20i 968	. . . . . 6 |- (A.x -. ph -> A.x(ph -> x = y))
5		19.8a 1005	. . . . . 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 1372	. . . 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 1374	. . . . 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 950  E.wex 956   = wceq 1099  E!weu 1357  E*wmo 1358

This theorem is referenced by:  mo3 1378  eu5 1386  immo 1394  moimv 1396  moanim 1404  mo2icl 1895  moabex 2734  dffun3 3468  dffunmof 3471  grothprim 8635

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360

Copyright terms: Public domain