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  849  dcfromcon  1468  dcfrompeirce  1469  nfbii  1496  sbi2v  1916  sbim  1981  sb8mo  2068  cbvmo  2094  necon4ddc  2448  raleqbii  2518  rmo5  2726  cbvrmo  2737  ss2ab  3261  snsssn  3802  trint  4158  ssextss  4265  ordsoexmid  4611  zfregfr  4623  tfi  4631  peano2  4644  peano5  4647  relop  4829  dmcosseq  4951  cotr  5065  issref  5066  cnvsym  5067  intasym  5068  intirr  5070  codir  5072  qfto  5073  cnvpom  5226  cnvsom  5227  funcnvuni  5344  poxp  6320  infmoti  7132  dfinfre  9031  bezoutlembi  12359  algcvgblem  12404  isprm2  12472  ntreq0  14637  ss1oel2o  15965