Theorem bibi1i 217

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 128	. 2 |- ( ( ph <-> ch ) <-> ( ch <-> ph ) )
2		bibi.a	. . 3 |- ( ph <-> ps )
3	2	bibi2i 216	. 2 |- ( ( ch <-> ph ) <-> ( ch <-> ps ) )
4		bicom 128	. 2 |- ( ( ch <-> ps ) <-> ( ps <-> ch ) )
5	1, 3, 4	3bitri 195	1 |- ( ( ph <-> ch ) <-> ( ps <-> ch ) )

Syntax hints:   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bibi12i  218  bilukdc  1287  sbrbis  1835  necon1abiddc  2267  necon1bbiddc  2268  necon4abiddc  2278  elrab3t  2697  ssequn1  3113  asymref  4710