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  2737  unass  3100  ssin  3159  difab  3206  iunss  3698  poirr  4044  cnvuni  4521  dfco2  4820  dff1o6  5416  elznn0  8260  bj-ssom  10060