Theorem sbbid 1834

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 1510	. 2 |- ( ph -> A. x ( ps <-> ch ) )
4		spsbbi 1832	. 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 104   A.wal 1341   F/wnf 1448   [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-nf 1449  df-sb 1751

This theorem is referenced by:  bezoutlemmain  11931