Theorem moimv 1412

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

Assertion

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

Distinct variable group:   ph,x

Proof of Theorem moimv

Step	Hyp	Ref	Expression
1		ax-1 4	. . . . . . 7 |- (ps -> (ph -> ps))
2	1	a1i 8	. . . . . 6 |- (ph -> (ps -> (ph -> ps)))
3	2	imim1d 28	. . . . 5 |- (ph -> (((ph -> ps) -> x = y) -> (ps -> x = y)))
4	3	19.20dv 1284	. . . 4 |- (ph -> (A.x((ph -> ps) -> x = y) -> A.x(ps -> x = y)))
5	4	19.22dv 1285	. . 3 |- (ph -> (E.yA.x((ph -> ps) -> x = y) -> E.yA.x(ps -> x = y)))
6		ax-17 968	. . . 4 |- ((ph -> ps) -> A.y(ph -> ps))
7	6	mo2 1393	. . 3 |- (E*x(ph -> ps) <-> E.yA.x((ph -> ps) -> x = y))
8		ax-17 968	. . . 4 |- (ps -> A.yps)
9	8	mo2 1393	. . 3 |- (E*xps <-> E.yA.x(ps -> x = y))
10	5, 7, 9	3imtr4g 551	. 2 |- (ph -> (E*x(ph -> ps) -> E*xps))
11	10	com12 11	1 |- (E*x(ph -> ps) -> (ph -> E*xps))

Syntax hints:   -> wi 3  A.wal 951  E.wex 977  E*wmo 1374

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376

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