Theorem bj-sbimeh 9912

Description: A strengthening of sbieh 1673 (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 1247	. . . 4 |- T.
2	1	hbth 1352	. . 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 9911	. 2 |- ( T. -> ( [ y / x ] ph -> ps ) )
8	7	trud 1252	1 |- ( [ y / x ] ph -> ps )

Syntax hints:   -> wi 4   A.wal 1241   T. wtru 1244   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-sb 1646

This theorem is referenced by:  bj-sbime  9913