Theorem spimth 1623

Description: Closed theorem form of spim 1626. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)

Assertion

Ref	Expression
spimth	|- ( A. x ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )

Proof of Theorem spimth

Step	Hyp	Ref	Expression
1		imim2 49	. . . . . 6 |- ( ( ps -> A. x ps ) -> ( ( ph -> ps ) -> ( ph -> A. x ps ) ) )
2	1	imim2d 48	. . . . 5 |- ( ( ps -> A. x ps ) -> ( ( x = y -> ( ph -> ps ) ) -> ( x = y -> ( ph -> A. x ps ) ) ) )
3	2	imp 115	. . . 4 |- ( ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) -> ( x = y -> ( ph -> A. x ps ) ) )
4	3	com23 72	. . 3 |- ( ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) -> ( ph -> ( x = y -> A. x ps ) ) )
5	4	al2imi 1347	. 2 |- ( A. x ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> A. x ( x = y -> A. x ps ) ) )
6		ax9o 1588	. 2 |- ( A. x ( x = y -> A. x ps ) -> ps )
7	5, 6	syl6 29	1 |- ( A. x ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  equveli  1642