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  563  sbanv  1936  sbexyz  2054  exists1  2174  euxfrdc  2989  euind  2990  rmo4  2996  rmo3f  3000  rmo3  3121  ddifstab  3336  opm  4319  uniuni  4541  rabxp  4755  eliunxp  4860  dmmrnm  4942  imadisj  5089  intirr  5114  resco  5232  funcnv3  5382  fncnv  5386  fun11  5387  fununi  5388  f1mpt  5894  mpomptx  6094  ixp0x  6871  mapsnen  6962  xpcomco  6981  enq0tr  7617  elq  9813  bitsmod  12462  pythagtrip  12801  ntreq0  14800  tx1cn  14937