Theorem moim 2119

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 1565	. . 3 |- F/ x A. x ( ph -> ps )
2		ax-4 1534	. . . . . 6 |- ( A. x ( ph -> ps ) -> ( ph -> ps ) )
3		spsbim 1867	. . . . . 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 1903	. . 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 1545	. 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 1550	. . 3 |- ( ps -> A. y ps )
9	8	mo3h 2108	. 2 |- ( E* x ps <-> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) )
10		ax-17 1550	. . 3 |- ( ph -> A. y ph )
11	10	mo3h 2108	. 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 1371   [wsb 1786   E*wmo 2056

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059

This theorem is referenced by:  moimi  2120  euimmo  2122  moexexdc  2139  euexex  2140  rmoim  2978  rmoimi2  2980  ssrmof  3260  disjss1  4033  reusv1  4513  funmo  5295  uptx  14821