Theorem spimth 1746

Description: Closed theorem form of spim 1749. (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 55	. . . . . 6 |- ( ( ps -> A. x ps ) -> ( ( ph -> ps ) -> ( ph -> A. x ps ) ) )
2	1	imim2d 54	. . . . 5 |- ( ( ps -> A. x ps ) -> ( ( x = y -> ( ph -> ps ) ) -> ( x = y -> ( ph -> A. x ps ) ) ) )
3	2	imp 124	. . . 4 |- ( ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) -> ( x = y -> ( ph -> A. x ps ) ) )
4	3	com23 78	. . 3 |- ( ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) -> ( ph -> ( x = y -> A. x ps ) ) )
5	4	al2imi 1469	. 2 |- ( A. x ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> A. x ( x = y -> A. x ps ) ) )
6		ax9o 1709	. 2 |- ( A. x ( x = y -> A. x ps ) -> ps )
7	5, 6	syl6 33	1 |- ( A. x ( ( ps -> A. x ps ) /\ ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  equveli  1770