Theorem bibi1i 228

Description: Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
bibi.a	|- ( ph <-> ps )

Assertion

Ref	Expression
bibi1i	|- ( ( ph <-> ch ) <-> ( ps <-> ch ) )

Proof of Theorem bibi1i

Step	Hyp	Ref	Expression
1		bicom 140	. 2 |- ( ( ph <-> ch ) <-> ( ch <-> ph ) )
2		bibi.a	. . 3 |- ( ph <-> ps )
3	2	bibi2i 227	. 2 |- ( ( ch <-> ph ) <-> ( ch <-> ps ) )
4		bicom 140	. 2 |- ( ( ch <-> ps ) <-> ( ps <-> ch ) )
5	1, 3, 4	3bitri 206	1 |- ( ( ph <-> ch ) <-> ( ps <-> ch ) )

Syntax hints:   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bibi12i  229  biadani  612  bilukdc  1396  sbrbis  1959  necon1abiddc  2407  necon1bbiddc  2408  necon4abiddc  2418  elrab3t  2890  ddifstab  3265  ssequn1  3303  asymref  5006