Theorem 3bitrri 207

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 185	. 2 |- ( ch <-> ph )
5	1, 4	bitr3i 186	1 |- ( th <-> ph )

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:  sbralie  2756  reu8  2969  unass  3330  ssin  3395  difab  3442  iunss  3968  poirr  4354  cnvuni  4864  dfco2  5182  dff1o6  5845  elznn0  9387  bj-ssom  15876