Theorem 3bitrri 196

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

Hypotheses

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

Assertion

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

Proof of Theorem 3bitrri

Step	Hyp	Ref	Expression
1		3bitri.3	. 2 |- (ch <-> th)
2		3bitri.1	. . 3 |- (ph <-> ps)
3		3bitri.2	. . 3 |- (ps <-> ch)
4	2, 3	bitr2i 174	. 2 |- (ch <-> ph)
5	1, 4	bitr3i 175	1 |- (th <-> ph)

Syntax hints:   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  reu8  2731  unass  3094  ssin  3153  difab  3200  iunss  3689  poirr  4035  cnvuni  4464  dfco2  4763  dff1o6  5359  elznn0  8036  bj-ssom  9395