Theorem morimv 2252

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

Assertion

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

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 6	. . . . . . 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 1621	. . . 4 |- (ph -> (A.x((ph -> ps) -> x = y) -> A.x(ps -> x = y)))
5	4	eximdv 1622	. . 3 |- (ph -> (E.yA.x((ph -> ps) -> x = y) -> E.yA.x(ps -> x = y)))
6		nfv 1619	. . . 4 |- F/y(ph -> ps)
7	6	mo2 2233	. . 3 |- (E*x(ph -> ps) <-> E.yA.x((ph -> ps) -> x = y))
8		nfv 1619	. . . 4 |- F/yps
9	8	mo2 2233	. . 3 |- (E*xps <-> E.yA.x(ps -> x = y))
10	5, 7, 9	3imtr4g 261	. 2 |- (ph -> (E*x(ph -> ps) -> E*xps))
11	10	com12 27	1 |- (E*x(ph -> ps) -> (ph -> E*xps))

Syntax hints:   -> wi 4  A.wal 1540  E.wex 1541   = wceq 1642  E*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by: (None)