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  457  oplem1  988  ax11i  1766  equsex  1780  eleq1a  2310  ceqsalg  2850  cgsexg  2857  cgsex2g  2858  cgsex4g  2859  ceqsex  2860  spc2egv  2915  spc3egv  2917  csbiebt  3187  dfiin2g  4043  sotricim  4466  ralxfrALT  4611  iunpw  4624  opelxp  4802  ssrel  4861  ssrel2  4863  ssrelrel  4873  iss  5107  funcnvuni  5448  fun11iun  5658  tfrlem8  6583  eroveu  6894  fundmen  7088  nneneq  7152  fidifsnen  7166  prarloclem5  7861  prarloc  7864  recexprlemss1l  7996  recexprlemss1u  7997  uzin  9938  indstr  9976  elfzmlbp  10522  swrdnd  11414  isclwwlknx  16640