Theorem moim 1964

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 1434	. . 3 |- F/ x A. x ( ph -> ps )
2		ax-4 1400	. . . . . 6 |- ( A. x ( ph -> ps ) -> ( ph -> ps ) )
3		spsbim 1724	. . . . . 6 |- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
4	2, 3	anim12d 318	. . . . 5 |- ( A. x ( ph -> ps ) -> ( ( ph /\ [ y / x ] ph ) -> ( ps /\ [ y / x ] ps ) ) )
5	4	imim1d 69	. . . 4 |- ( A. x ( ph -> ps ) -> ( ( ( ps /\ [ y / x ] ps ) -> x = y ) -> ( ( ph /\ [ y / x ] ph ) -> x = y ) ) )
6	5	alimdv 1759	. . 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 1414	. 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 1419	. . 3 |- ( ps -> A. y ps )
9	8	mo3h 1953	. 2 |- ( E* x ps <-> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) )
10		ax-17 1419	. . 3 |- ( ph -> A. y ph )
11	10	mo3h 1953	. 2 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
12	7, 9, 11	3imtr4g 194	1 |- ( A. x ( ph -> ps ) -> ( E* x ps -> E* x ph ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   [wsb 1645   E*wmo 1901

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904

This theorem is referenced by:  moimi  1965  euimmo  1967  moexexdc  1984  euexex  1985  rmoim  2740  rmoimi2  2742  disjss1  3751  reusv1  4190  funmo  4917