Theorem spimt 1664

Description: Closed theorem form of spim 1666. (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 1626	. . . 4 |- E. x x = y
2		exim 1530	. . . 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 1555	. . 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 1573	. . 3 |- ( F/ x ps -> ( E. x ps <-> ps ) )
7	6	biimpd 142	. 2 |- ( F/ x ps -> ( E. x ps -> ps ) )
8	5, 7	sylan9r 402	1 |- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1282   = wceq 1284   F/wnf 1389   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390

This theorem is referenced by:  spimd  10576