Theorem bibi1i 226

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 138	. 2 |- ( ( ph <-> ch ) <-> ( ch <-> ph ) )
2		bibi.a	. . 3 |- ( ph <-> ps )
3	2	bibi2i 225	. 2 |- ( ( ch <-> ph ) <-> ( ch <-> ps ) )
4		bicom 138	. 2 |- ( ( ch <-> ps ) <-> ( ps <-> ch ) )
5	1, 3, 4	3bitri 204	1 |- ( ( ph <-> ch ) <-> ( ps <-> ch ) )

Syntax hints:   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  bibi12i  227  bilukdc  1332  sbrbis  1883  necon1abiddc  2317  necon1bbiddc  2318  necon4abiddc  2328  elrab3t  2770  ddifstab  3132  ssequn1  3170  asymref  4817