Theorem mopick 2078

Description: "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)

Assertion

Ref	Expression
mopick	|- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )

Proof of Theorem mopick

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1507	. . . 4 |- ( ( ph /\ ps ) -> A. y ( ph /\ ps ) )
2		hbs1 1912	. . . . 5 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3		hbs1 1912	. . . . 5 |- ( [ y / x ] ps -> A. x [ y / x ] ps )
4	2, 3	hban 1527	. . . 4 |- ( ( [ y / x ] ph /\ [ y / x ] ps ) -> A. x ( [ y / x ] ph /\ [ y / x ] ps ) )
5		sbequ12 1745	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
6		sbequ12 1745	. . . . 5 |- ( x = y -> ( ps <-> [ y / x ] ps ) )
7	5, 6	anbi12d 465	. . . 4 |- ( x = y -> ( ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) ) )
8	1, 4, 7	cbvexh 1729	. . 3 |- ( E. x ( ph /\ ps ) <-> E. y ( [ y / x ] ph /\ [ y / x ] ps ) )
9		ax-17 1507	. . . . . . 7 |- ( ph -> A. y ph )
10	9	mo3h 2053	. . . . . 6 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
11		ax-4 1488	. . . . . . 7 |- ( A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ( ph /\ [ y / x ] ph ) -> x = y ) )
12	11	sps 1518	. . . . . 6 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ( ph /\ [ y / x ] ph ) -> x = y ) )
13	10, 12	sylbi 120	. . . . 5 |- ( E* x ph -> ( ( ph /\ [ y / x ] ph ) -> x = y ) )
14		sbequ2 1743	. . . . . . . . 9 |- ( x = y -> ( [ y / x ] ps -> ps ) )
15	14	imim2i 12	. . . . . . . 8 |- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ( ph /\ [ y / x ] ph ) -> ( [ y / x ] ps -> ps ) ) )
16	15	expd 256	. . . . . . 7 |- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ph -> ( [ y / x ] ph -> ( [ y / x ] ps -> ps ) ) ) )
17	16	com4t 85	. . . . . 6 |- ( [ y / x ] ph -> ( [ y / x ] ps -> ( ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ph -> ps ) ) ) )
18	17	imp 123	. . . . 5 |- ( ( [ y / x ] ph /\ [ y / x ] ps ) -> ( ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ph -> ps ) ) )
19	13, 18	syl5 32	. . . 4 |- ( ( [ y / x ] ph /\ [ y / x ] ps ) -> ( E* x ph -> ( ph -> ps ) ) )
20	19	exlimiv 1578	. . 3 |- ( E. y ( [ y / x ] ph /\ [ y / x ] ps ) -> ( E* x ph -> ( ph -> ps ) ) )
21	8, 20	sylbi 120	. 2 |- ( E. x ( ph /\ ps ) -> ( E* x ph -> ( ph -> ps ) ) )
22	21	impcom 124	1 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1330   E.wex 1469   [wsb 1736   E*wmo 2001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004

This theorem is referenced by:  eupick  2079  mopick2  2083  moexexdc  2084  euexex  2085  morex  2872  imadif  5211  funimaexglem  5214