Theorem 3bitr2i 208

Description: A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)

Hypotheses

Ref	Expression
3bitr2i.1	|- ( ph <-> ps )
3bitr2i.2	|- ( ch <-> ps )
3bitr2i.3	|- ( ch <-> th )

Assertion

Ref	Expression
3bitr2i	|- ( ph <-> th )

Proof of Theorem 3bitr2i

Step	Hyp	Ref	Expression
1		3bitr2i.1	. . 3 |- ( ph <-> ps )
2		3bitr2i.2	. . 3 |- ( ch <-> ps )
3	1, 2	bitr4i 187	. 2 |- ( ph <-> ch )
4		3bitr2i.3	. 2 |- ( ch <-> th )
5	3, 4	bitri 184	1 |- ( ph <-> th )

Syntax hints:   <-> 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:  an13  569  sbanv  1944  sbexyz  2063  exists1  2183  euxfrdc  3012  euind  3013  rmo4  3019  rmo3f  3023  rmo3  3144  ddifstab  3361  opm  4369  uniuni  4592  rabxp  4807  eliunxp  4914  dmmrnm  4996  imadisj  5144  intirr  5169  resco  5287  funcnv3  5438  fncnv  5442  fun11  5443  fununi  5444  f1mpt  5967  mpomptx  6169  ixp0x  6998  mapsnen  7090  xpcomco  7114  enq0tr  7791  elq  10001  bitsmod  12701  pythagtrip  13040  ntreq0  15156  tx1cn  15293