Theorem mopick 2019

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 1459	. . . 4 |- ( ( ph /\ ps ) -> A. y ( ph /\ ps ) )
2		hbs1 1855	. . . . 5 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3		hbs1 1855	. . . . 5 |- ( [ y / x ] ps -> A. x [ y / x ] ps )
4	2, 3	hban 1479	. . . 4 |- ( ( [ y / x ] ph /\ [ y / x ] ps ) -> A. x ( [ y / x ] ph /\ [ y / x ] ps ) )
5		sbequ12 1694	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
6		sbequ12 1694	. . . . 5 |- ( x = y -> ( ps <-> [ y / x ] ps ) )
7	5, 6	anbi12d 456	. . . 4 |- ( x = y -> ( ( ph /\ ps ) <-> ( [ y / x ] ph /\ [ y / x ] ps ) ) )
8	1, 4, 7	cbvexh 1678	. . 3 |- ( E. x ( ph /\ ps ) <-> E. y ( [ y / x ] ph /\ [ y / x ] ps ) )
9		ax-17 1459	. . . . . . 7 |- ( ph -> A. y ph )
10	9	mo3h 1994	. . . . . 6 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
11		ax-4 1440	. . . . . . 7 |- ( A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ( ph /\ [ y / x ] ph ) -> x = y ) )
12	11	sps 1470	. . . . . 6 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ( ph /\ [ y / x ] ph ) -> x = y ) )
13	10, 12	sylbi 119	. . . . 5 |- ( E* x ph -> ( ( ph /\ [ y / x ] ph ) -> x = y ) )
14		sbequ2 1692	. . . . . . . . 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 254	. . . . . . 7 |- ( ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ph -> ( [ y / x ] ph -> ( [ y / x ] ps -> ps ) ) ) )
17	16	com4t 84	. . . . . 6 |- ( [ y / x ] ph -> ( [ y / x ] ps -> ( ( ( ph /\ [ y / x ] ph ) -> x = y ) -> ( ph -> ps ) ) ) )
18	17	imp 122	. . . . 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 1529	. . 3 |- ( E. y ( [ y / x ] ph /\ [ y / x ] ps ) -> ( E* x ph -> ( ph -> ps ) ) )
21	8, 20	sylbi 119	. 2 |- ( E. x ( ph /\ ps ) -> ( E* x ph -> ( ph -> ps ) ) )
22	21	impcom 123	1 |- ( ( E* x ph /\ E. x ( ph /\ ps ) ) -> ( ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   E.wex 1421   [wsb 1685   E*wmo 1942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945

This theorem is referenced by:  eupick  2020  mopick2  2024  moexexdc  2025  euexex  2026  morex  2776  imadif  4999  funimaexglem  5002