Theorem spsbim 1831

Description: Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)

Assertion

Ref	Expression
spsbim	|- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )

Proof of Theorem spsbim

Step	Hyp	Ref	Expression
1		imim2 55	. . . 4 |- ( ( ph -> ps ) -> ( ( x = y -> ph ) -> ( x = y -> ps ) ) )
2	1	sps 1525	. . 3 |- ( A. x ( ph -> ps ) -> ( ( x = y -> ph ) -> ( x = y -> ps ) ) )
3		id 19	. . . . . 6 |- ( ( ph -> ps ) -> ( ph -> ps ) )
4	3	anim2d 335	. . . . 5 |- ( ( ph -> ps ) -> ( ( x = y /\ ph ) -> ( x = y /\ ps ) ) )
5	4	alimi 1443	. . . 4 |- ( A. x ( ph -> ps ) -> A. x ( ( x = y /\ ph ) -> ( x = y /\ ps ) ) )
6		exim 1587	. . . 4 |- ( A. x ( ( x = y /\ ph ) -> ( x = y /\ ps ) ) -> ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ ps ) ) )
7	5, 6	syl 14	. . 3 |- ( A. x ( ph -> ps ) -> ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ ps ) ) )
8	2, 7	anim12d 333	. 2 |- ( A. x ( ph -> ps ) -> ( ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) -> ( ( x = y -> ps ) /\ E. x ( x = y /\ ps ) ) ) )
9		df-sb 1751	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
10		df-sb 1751	. 2 |- ( [ y / x ] ps <-> ( ( x = y -> ps ) /\ E. x ( x = y /\ ps ) ) )
11	8, 9, 10	3imtr4g 204	1 |- ( A. x ( ph -> ps ) -> ( [ y / x ] ph -> [ y / x ] ps ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   E.wex 1480   [wsb 1750

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-sb 1751

This theorem is referenced by:  spsbbi  1832  hbsb4t  2001  moim  2078