Theorem moim 2078

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 1529	. . 3 |- F/ x A. x ( ph -> ps )
2		ax-4 1498	. . . . . 6 |- ( A. x ( ph -> ps ) -> ( ph -> ps ) )
3		spsbim 1831	. . . . . 6 |- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
4	2, 3	anim12d 333	. . . . 5 |- ( A. x ( ph -> ps ) -> ( ( ph /\ [ y / x ] ph ) -> ( ps /\ [ y / x ] ps ) ) )
5	4	imim1d 75	. . . 4 |- ( A. x ( ph -> ps ) -> ( ( ( ps /\ [ y / x ] ps ) -> x = y ) -> ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
6	5	alimdv 1867	. . 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 1509	. 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 1514	. . 3 |- ( ps -> A. y ps )
9	8	mo3h 2067	. 2 |- ( E* x ps <-> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) )
10		ax-17 1514	. . 3 |- ( ph -> A. y ph )
11	10	mo3h 2067	. 2 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
12	7, 9, 11	3imtr4g 204	1 |- ( A. x ( ph -> ps ) -> ( E* x ps -> E* x ph ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   [wsb 1750   E*wmo 2015

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018

This theorem is referenced by:  moimi  2079  euimmo  2081  moexexdc  2098  euexex  2099  rmoim  2927  rmoimi2  2929  ssrmof  3205  disjss1  3965  reusv1  4436  funmo  5203  uptx  12914