Theorem biimprcd 160

Description: Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)

Hypothesis

Ref	Expression
biimpcd.1	|- ( ph -> ( ps <-> ch ) )

Assertion

Ref	Expression
biimprcd	|- ( ch -> ( ph -> ps ) )

Proof of Theorem biimprcd

Step	Hyp	Ref	Expression
1		id 19	. 2 |- ( ch -> ch )
2		biimpcd.1	. 2 |- ( ph -> ( ps <-> ch ) )
3	1, 2	syl5ibrcom 157	1 |- ( ch -> ( ph -> ps ) )

Syntax hints:   -> wi 4   <-> 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:  biimparc  299  pm5.32  453  oplem1  981  ax11i  1760  equsex  1774  eleq1a  2301  ceqsalg  2828  cgsexg  2835  cgsex2g  2836  cgsex4g  2837  ceqsex  2838  spc2egv  2893  spc3egv  2895  csbiebt  3164  dfiin2g  3998  sotricim  4415  ralxfrALT  4559  iunpw  4572  opelxp  4750  ssrel  4809  ssrel2  4811  ssrelrel  4821  iss  5054  funcnvuni  5393  fun11iun  5598  tfrlem8  6475  eroveu  6786  fundmen  6972  nneneq  7031  fidifsnen  7045  prarloclem5  7703  prarloc  7706  recexprlemss1l  7838  recexprlemss1u  7839  uzin  9772  indstr  9805  elfzmlbp  10345  swrdnd  11212  isclwwlknx  16184