Theorem bj-sbimeh 10583

Description: A strengthening of sbieh 1713 (same proof). (Contributed by BJ, 16-Dec-2019.)

Hypotheses

Ref	Expression
bj-sbimeh.1	|- ( ps -> A. x ps )
bj-sbimeh.2	|- ( x = y -> ( ph -> ps ) )

Assertion

Ref	Expression
bj-sbimeh	|- ( [ y / x ] ph -> ps )

Proof of Theorem bj-sbimeh

Step	Hyp	Ref	Expression
1		tru 1288	. . . 4 |- T.
2	1	hbth 1392	. . 3 |- ( T. -> A. x T. )
3		bj-sbimeh.1	. . . 4 |- ( ps -> A. x ps )
4	3	a1i 9	. . 3 |- ( T. -> ( ps -> A. x ps ) )
5		bj-sbimeh.2	. . . 4 |- ( x = y -> ( ph -> ps ) )
6	5	a1i 9	. . 3 |- ( T. -> ( x = y -> ( ph -> ps ) ) )
7	2, 4, 6	bj-sbimedh 10582	. 2 |- ( T. -> ( [ y / x ] ph -> ps ) )
8	7	trud 1293	1 |- ( [ y / x ] ph -> ps )

Syntax hints:   -> wi 4   A.wal 1282   T. wtru 1285   [wsb 1685

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-ial 1467

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-sb 1686

This theorem is referenced by:  bj-sbime  10584