Theorem moim 2006

Description: "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)

Assertion

Ref	Expression
moim	|- ( A. x ( ph -> ps ) -> ( E* x ps -> E* x ph ) )

Proof of Theorem moim

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfa1 1475	. . 3 |- F/ x A. x ( ph -> ps )
2		ax-4 1441	. . . . . 6 |- ( A. x ( ph -> ps ) -> ( ph -> ps ) )
3		spsbim 1765	. . . . . 6 |- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
4	2, 3	anim12d 328	. . . . 5 |- ( A. x ( ph -> ps ) -> ( ( ph /\ [ y / x ] ph ) -> ( ps /\ [ y / x ] ps ) ) )
5	4	imim1d 74	. . . 4 |- ( A. x ( ph -> ps ) -> ( ( ( ps /\ [ y / x ] ps ) -> x = y ) -> ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
6	5	alimdv 1801	. . 3 |- ( A. x ( ph -> ps ) -> ( A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) -> A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
7	1, 6	alimd 1455	. 2 |- ( A. x ( ph -> ps ) -> ( A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
8		ax-17 1460	. . 3 |- ( ps -> A. y ps )
9	8	mo3h 1995	. 2 |- ( E* x ps <-> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) )
10		ax-17 1460	. . 3 |- ( ph -> A. y ph )
11	10	mo3h 1995	. 2 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
12	7, 9, 11	3imtr4g 203	1 |- ( A. x ( ph -> ps ) -> ( E* x ps -> E* x ph ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1283   [wsb 1686   E*wmo 1943

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946

This theorem is referenced by:  moimi  2007  euimmo  2009  moexexdc  2026  euexex  2027  rmoim  2792  rmoimi2  2794  disjss1  3774  reusv1  4210  funmo  4941