Theorem bibi2i 226

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	syl6bb 195	. 2 |- ( ( ch <-> ph ) -> ( ch <-> ps ) )
4		id 19	. . 3 |- ( ( ch <-> ps ) -> ( ch <-> ps ) )
5	4, 2	syl6bbr 197	. 2 |- ( ( ch <-> ps ) -> ( ch <-> ph ) )
6	3, 5	impbii 125	1 |- ( ( ch <-> ph ) <-> ( ch <-> ps ) )

Syntax hints:   <-> wb 104

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  bibi1i  227  bibi12i  228  bibi2d  231  pm4.71r  385  sblbis  1909  sbrbif  1911  abeq2  2223  abid2f  2280  necon4biddc  2357  pm13.183  2792  disj3  3381  euabsn2  3558  a9evsep  4010  inex1  4022  zfpair2  4092  sucel  4292  bdinex1  12789  bj-zfpair2  12800  bj-d0clsepcl  12815