Theorem bibi2i 227

Description: Inference adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.)

Hypothesis

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

Assertion

Ref	Expression
bibi2i	|- ( ( ch <-> ph ) <-> ( ch <-> ps ) )

Proof of Theorem bibi2i

Step	Hyp	Ref	Expression
1		id 19	. . 3 |- ( ( ch <-> ph ) -> ( ch <-> ph ) )
2		bibi.a	. . 3 |- ( ph <-> ps )
3	1, 2	bitrdi 196	. 2 |- ( ( ch <-> ph ) -> ( ch <-> ps ) )
4		id 19	. . 3 |- ( ( ch <-> ps ) -> ( ch <-> ps ) )
5	4, 2	bitr4di 198	. 2 |- ( ( ch <-> ps ) -> ( ch <-> ph ) )
6	3, 5	impbii 126	1 |- ( ( ch <-> ph ) <-> ( ch <-> ps ) )

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:  bibi1i  228  bibi12i  229  bibi2d  232  pm4.71r  390  sblbis  1960  sbrbif  1962  abeq2  2286  abid2f  2345  necon4biddc  2422  pm13.183  2875  disj3  3475  euabsn2  3661  a9evsep  4125  inex1  4137  zfpair2  4210  sucel  4410  bdinex1  14621  bj-zfpair2  14632  bj-d0clsepcl  14647