Theorem spimt 1954

Description: Closed theorem form of spim 1955. (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		ax6e 1951	. . . 4 |- E. x x = y
2		exim 1628	. . . 4 |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( E. x x = y -> E. x ( ph -> ps ) ) )
3	1, 2	mpi 17	. . 3 |- ( A. x ( x = y -> ( ph -> ps ) ) -> E. x ( ph -> ps ) )
4		19.35 1659	. . 3 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )
5	3, 4	sylib 196	. 2 |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( A. x ph -> E. x ps ) )
6		19.9t 1828	. . 3 |- ( F/ x ps -> ( E. x ps <-> ps ) )
7	6	biimpd 207	. 2 |- ( F/ x ps -> ( E. x ps -> ps ) )
8	5, 7	sylan9r 653	1 |- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 369   A.wal 1362   E.wex 1591   F/wnf 1594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-12 1797  ax-13 1948

This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-nf 1595

This theorem is referenced by: (None)