Theorem moimv 2026

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 5	. . . . . . 7 |- ( ps -> ( ph -> ps ) )
2	1	a1i 9	. . . . . 6 |- ( ph -> ( ps -> ( ph -> ps ) ) )
3	2	sbimi 1705	. . . . . . 7 |- ( [ y / x ] ph -> [ y / x ] ( ps -> ( ph -> ps ) ) )
4		nfv 1476	. . . . . . . 8 |- F/ x ph
5	4	sbf 1718	. . . . . . 7 |- ( [ y / x ] ph <-> ph )
6		sbim 1887	. . . . . . 7 |- ( [ y / x ] ( ps -> ( ph -> ps ) ) <-> ( [ y / x ] ps -> [ y / x ] ( ph -> ps ) ) )
7	3, 5, 6	3imtr3i 199	. . . . . 6 |- ( ph -> ( [ y / x ] ps -> [ y / x ] ( ph -> ps ) ) )
8	2, 7	anim12d 331	. . . . 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 1820	. . 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 1474	. . . 4 |- ( ( ph -> ps ) -> A. y ( ph -> ps ) )
12	11	mo3h 2013	. . 3 |- ( E* x ( ph -> ps ) <-> A. x A. y ( ( ( ph -> ps ) /\ [ y / x ] ( ph -> ps ) ) -> x = y ) )
13		ax-17 1474	. . . 4 |- ( ps -> A. y ps )
14	13	mo3h 2013	. . 3 |- ( E* x ps <-> A. x A. y ( ( ps /\ [ y / x ] ps ) -> x = y ) )
15	10, 12, 14	3imtr4g 204	. 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 103   A.wal 1297   [wsb 1703   E*wmo 1961

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964

This theorem is referenced by: (None)