Theorem imbi12i 239

Description: Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
imbi12i.1	|- ( ph <-> ps )
imbi12i.2	|- ( ch <-> th )

Assertion

Ref	Expression
imbi12i	|- ( ( ph -> ch ) <-> ( ps -> th ) )

Proof of Theorem imbi12i

Step	Hyp	Ref	Expression
1		imbi12i.2	. . 3 |- ( ch <-> th )
2	1	imbi2i 226	. 2 |- ( ( ph -> ch ) <-> ( ph -> th ) )
3		imbi12i.1	. . 3 |- ( ph <-> ps )
4	3	imbi1i 238	. 2 |- ( ( ph -> th ) <-> ( ps -> th ) )
5	2, 4	bitri 184	1 |- ( ( ph -> ch ) <-> ( ps -> th ) )

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:  stdcn  855  dcfromcon  1494  dcfrompeirce  1495  nfbii  1522  sbi2v  1943  sbim  2009  sb8mo  2096  cbvmo  2122  necon4ddc  2486  raleqbii  2556  rmo5  2767  cbvrmo  2779  ss2ab  3310  snsssn  3871  trint  4229  ssextss  4342  ordsoexmid  4690  zfregfr  4702  tfi  4710  peano2  4723  peano5  4726  relop  4911  dmcosseq  5035  cotr  5150  issref  5151  cnvsym  5152  intasym  5153  intirr  5155  codir  5157  qfto  5158  cnvpom  5311  cnvsom  5312  funcnvuni  5431  poxp  6442  infmoti  7333  dfinfre  9251  bezoutlembi  12731  algcvgblem  12776  isprm2  12844  ntreq0  15128  ss1oel2o  16902