Theorem biimpr 130

Description: Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)

Assertion

Ref	Expression
biimpr	|- ( ( ph <-> ps ) -> ( ps -> ph ) )

Proof of Theorem biimpr

Step	Hyp	Ref	Expression
1		df-bi 117	. . 3 |- ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) /\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) )
2	1	simpli 111	. 2 |- ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
3	2	simprd 114	1 |- ( ( ph <-> ps ) -> ( ps -> ph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105

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

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bicom1  131  pm5.74  179  bi3ant  224  pm5.32d  450  notbi  667  nbn2  698  pm4.72  828  con4biddc  858  con1biimdc  874  bijadc  883  pclem6  1385  exbir  1447  simplbi2comg  1454  albi  1482  exbi  1618  equsexd  1743  cbv2h  1762  cbv2w  1764  sbiedh  1801  ceqsalt  2789  spcegft  2843  elab3gf  2914  euind  2951  reu6  2953  reuind  2969  sbciegft  3020  iota4  5238  fv3  5581  algcvgblem  12217  bj-inf2vnlem1  15616