Theorem moimv 1044

Description: Move antecedent outside of "at most one".

Assertion

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

Distinct variable group(s):   ph,x

Proof of Theorem moimv

Step	Hyp	Ref	Expression
1		ax-1 3	. . . . . . 7 |- (ps -> (ph -> ps))
2	1	a1i 7	. . . . . 6 |- (ph -> (ps -> (ph -> ps)))
3	2	syl4d 28	. . . . 5 |- (ph -> (((ph -> ps) -> x = y) -> (ps -> x = y)))
4	3	19.20dv 946	. . . 4 |- (ph -> (A.x((ph -> ps) -> x = y) -> A.x(ps -> x = y)))
5	4	19.22dv 947	. . 3 |- (ph -> (E.yA.x((ph -> ps) -> x = y) -> E.yA.x(ps -> x = y)))
6		ax-17 925	. . . 4 |- ((ph -> ps) -> A.y(ph -> ps))
7	6	mo2 1026	. . 3 |- (E*x(ph -> ps) <-> E.yA.x((ph -> ps) -> x = y))
8		ax-17 925	. . . 4 |- (ps -> A.yps)
9	8	mo2 1026	. . 3 |- (E*xps <-> E.yA.x(ps -> x = y))
10	5, 7, 9	3imtr4g 426	. 2 |- (ph -> (E*x(ph -> ps) -> E*xps))
11	10	com12 13	1 |- (E*x(ph -> ps) -> (ph -> E*xps))

Syntax hints:   -> wi 2  A.wal 672  E.wex 678   = weq 797  E*wmo 1008

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

metamath.org