Theorem morimv 2191

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

Assertion

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

Distinct variable group:   ph, x

Allowed substitution hint:   ps( x)

Proof of Theorem morimv

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1 5	. . . . . . 7 |- ( ps -> ( ph -> ps ) )
2	1	a1i 10	. . . . . 6 |- ( ph -> ( ps -> ( ph -> ps ) ) )
3	2	imim1d 69	. . . . 5 |- ( ph -> ( ( ( ph -> ps ) -> x = y ) -> ( ps -> x = y ) ) )
4	3	alimdv 1607	. . . 4 |- ( ph -> ( A. x ( ( ph -> ps ) -> x = y ) -> A. x ( ps -> x = y ) ) )
5	4	eximdv 1608	. . 3 |- ( ph -> ( E. y A. x ( ( ph -> ps ) -> x = y ) -> E. y A. x ( ps -> x = y ) ) )
6		nfv 1605	. . . 4 |- F/ y ( ph -> ps )
7	6	mo2 2172	. . 3 |- ( E* x ( ph -> ps ) <-> E. y A. x ( ( ph -> ps ) -> x = y ) )
8		nfv 1605	. . . 4 |- F/ y ps
9	8	mo2 2172	. . 3 |- ( E* x ps <-> E. y A. x ( ps -> x = y ) )
10	5, 7, 9	3imtr4g 261	. 2 |- ( ph -> ( E* x ( ph -> ps ) -> E* x ps ) )
11	10	com12 27	1 |- ( E* x ( ph -> ps ) -> ( ph -> E* x ps ) )

Syntax hints:   -> wi 4   A.wal 1527   E.wex 1528   = wceq 1623   E*wmo 2144

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148