Theorem sbbid 1726

Description: Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
sbbid.1	|- ( ph -> A. x ph )
sbbid.2	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
sbbid	|- ( ph -> ( [ y / x ] ps <-> [ y / x ] ch ) )

Proof of Theorem sbbid

Step	Hyp	Ref	Expression
1		sbbid.1	. . 3 |- ( ph -> A. x ph )
2		sbbid.2	. . 3 |- ( ph -> ( ps <-> ch ) )
3	1, 2	alrimih 1358	. 2 |- ( ph -> A. x ( ps <-> ch ) )
4		spsbbi 1725	. 2 |- ( A. x ( ps <-> ch ) -> ( [ y / x ] ps <-> [ y / x ] ch ) )
5	3, 4	syl 14	1 |- ( ph -> ( [ y / x ] ps <-> [ y / x ] ch ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   [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-sb 1646

This theorem is referenced by:  sbcomxyyz  1846