Theorem moim 2109

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 1555	. . 3 |- F/ x A. x ( ph -> ps )
2		ax-4 1524	. . . . . 6 |- ( A. x ( ph -> ps ) -> ( ph -> ps ) )
3		spsbim 1857	. . . . . 6 |- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )
4	2, 3	anim12d 335	. . . . 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 1893	. . 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 1535	. 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 1540	. . 3 |- ( ps -> A. y ps )
9	8	mo3h 2098	. 2 |- ( E* x ps <-> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) )
10		ax-17 1540	. . 3 |- ( ph -> A. y ph )
11	10	mo3h 2098	. 2 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
12	7, 9, 11	3imtr4g 205	1 |- ( A. x ( ph -> ps ) -> ( E* x ps -> E* x ph ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   [wsb 1776   E*wmo 2046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049

This theorem is referenced by:  moimi  2110  euimmo  2112  moexexdc  2129  euexex  2130  rmoim  2965  rmoimi2  2967  ssrmof  3246  disjss1  4016  reusv1  4493  funmo  5273  uptx  14510