Theorem sbbid 2403

Description: Deduction substituting both sides of a biconditional. (Contributed by NM, 30-Jun-1993.)

Hypotheses

Ref	Expression
sbbid.1	|- F/ 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 |- F/ x ph
2		sbbid.2	. . 3 |- ( ph -> ( ps <-> ch ) )
3	1, 2	alrimi 2082	. 2 |- ( ph -> A. x ( ps <-> ch ) )
4		spsbbi 2402	. 2 |- ( A. x ( ps <-> ch ) -> ( [ y / x ] ps <-> [ y / x ] ch ) )
5	3, 4	syl 17	1 |- ( ph -> ( [ y / x ] ps <-> [ y / x ] ch ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   F/wnf 1708   [wsb 1880

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-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbcom3  2411  sbco3  2417  sbcom2  2445  sbal  2462  wl-equsb3  33337  wl-sbcom2d-lem1  33342  wl-sbcom3  33372