Theorem moimv 2111

Description: Move antecedent outside of "at most one". (Contributed by NM, 28-Jul-1995.)

Assertion

Ref	Expression
moimv	|- ( E* x ( ph -> ps ) -> ( ph -> E* x ps ) )

Distinct variable group:   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem moimv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . . . 7 |- ( ps -> ( ph -> ps ) )
2	1	a1i 9	. . . . . 6 |- ( ph -> ( ps -> ( ph -> ps ) ) )
3	2	sbimi 1778	. . . . . . 7 |- ( [ y / x ] ph -> [ y / x ] ( ps -> ( ph -> ps ) ) )
4		nfv 1542	. . . . . . . 8 |- F/ x ph
5	4	sbf 1791	. . . . . . 7 |- ( [ y / x ] ph <-> ph )
6		sbim 1972	. . . . . . 7 |- ( [ y / x ] ( ps -> ( ph -> ps ) ) <-> ( [ y / x ] ps -> [ y / x ] ( ph -> ps ) ) )
7	3, 5, 6	3imtr3i 200	. . . . . 6 |- ( ph -> ( [ y / x ] ps -> [ y / x ] ( ph -> ps ) ) )
8	2, 7	anim12d 335	. . . . 5 |- ( ph -> ( ( ps /\ [ y / x ] ps ) -> ( ( ph -> ps ) /\ [ y / x ] ( ph -> ps ) ) ) )
9	8	imim1d 75	. . . 4 |- ( ph -> ( ( ( ( ph -> ps ) /\ [ y / x ] ( ph -> ps ) ) -> x = y ) -> ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
10	9	2alimdv 1895	. . 3 |- ( ph -> ( A. x A. y ( ( ( ph -> ps ) /\ [ y / x ] ( ph -> ps ) ) -> x = y ) -> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) ) )
11		ax-17 1540	. . . 4 |- ( ( ph -> ps ) -> A. y ( ph -> ps ) )
12	11	mo3h 2098	. . 3 |- ( E* x ( ph -> ps ) <-> A. x A. y ( ( ( ph -> ps ) /\ [ y / x ] ( ph -> ps ) ) -> x = y ) )
13		ax-17 1540	. . . 4 |- ( ps -> A. y ps )
14	13	mo3h 2098	. . 3 |- ( E* x ps <-> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) )
15	10, 12, 14	3imtr4g 205	. 2 |- ( ph -> ( E* x ( ph -> ps ) -> E* x ps ) )
16	15	com12 30	1 |- ( E* x ( ph -> ps ) -> ( ph -> E* x ps ) )

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: (None)