Theorem sbrbif 1950

Description: Introduce right biconditional inside of a substitution. (Contributed by NM, 18-Aug-1993.)

Hypotheses

Ref	Expression
sbrbif.1	|- ( ch -> A. x ch )
sbrbif.2	|- ( [ y / x ] ph <-> ps )

Assertion

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

Proof of Theorem sbrbif

Step	Hyp	Ref	Expression
1		sbrbif.2	. . 3 |- ( [ y / x ] ph <-> ps )
2	1	sbrbis 1949	. 2 |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> [ y / x ] ch ) )
3		sbrbif.1	. . . 4 |- ( ch -> A. x ch )
4	3	sbh 1764	. . 3 |- ( [ y / x ] ch <-> ch )
5	4	bibi2i 226	. 2 |- ( ( ps <-> [ y / x ] ch ) <-> ( ps <-> ch ) )
6	2, 5	bitri 183	1 |- ( [ y / x ] ( ph <-> ch ) <-> ( ps <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   [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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by: (None)