Theorem spimt 2253

Description: Closed theorem form of spim 2254. (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 2250	. . . 4 |- E. x x = y
2		exim 1761	. . . 4 |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( E. x x = y -> E. x ( ph -> ps ) ) )
3	1, 2	mpi 20	. . 3 |- ( A. x ( x = y -> ( ph -> ps ) ) -> E. x ( ph -> ps ) )
4		19.35 1805	. . 3 |- ( E. x ( ph -> ps ) <-> ( A. x ph -> E. x ps ) )
5	3, 4	sylib 208	. 2 |- ( A. x ( x = y -> ( ph -> ps ) ) -> ( A. x ph -> E. x ps ) )
6		19.9t 2071	. . 3 |- ( F/ x ps -> ( E. x ps <-> ps ) )
7	6	biimpd 219	. 2 |- ( F/ x ps -> ( E. x ps -> ps ) )
8	5, 7	sylan9r 690	1 |- ( ( F/ x ps /\ A. x ( x = y -> ( ph -> ps ) ) ) -> ( A. x ph -> ps ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)