Theorem spimt 1747

Description: Closed theorem form of spim 1749. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)

Assertion

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

Proof of Theorem spimt

Step	Hyp	Ref	Expression
1		a9e 1707	. . . 4 |- E. x x = y
2		exim 1610	. . . 4 |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( E. x x = y -> E. x ( ph -> ps ) ) )
3	1, 2	mpi 15	. . 3 |- ( A. x ( x = y -> ( ph -> ps ) ) -> E. x ( ph -> ps ) )
4		19.35-1 1635	. . 3 |- ( E. x ( ph -> ps ) -> ( A. x ph -> E. x ps ) )
5	3, 4	syl 14	. 2 |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( A. x ph -> E. x ps ) )
6		19.9t 1653	. . 3 |- ( F/ x ps -> ( E. x ps <-> ps ) )
7	6	biimpd 144	. 2 |- ( F/ x ps -> ( E. x ps -> ps ) )
8	5, 7	sylan9r 410	1 |- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   F/wnf 1471   E.wex 1503

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  df-nf 1472

This theorem is referenced by:  spimd  15257