Theorem spimt 1729

Description: Closed theorem form of spim 1731. (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 1689	. . . 4 |- E. x x = y
2		exim 1592	. . . 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 1617	. . 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 1635	. . 3 |- ( F/ x ps -> ( E. x ps <-> ps ) )
7	6	biimpd 143	. 2 |- ( F/ x ps -> ( E. x ps -> ps ) )
8	5, 7	sylan9r 408	1 |- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   = wceq 1348   F/wnf 1453   E.wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454

This theorem is referenced by:  spimd  13800