Theorem 3bitr3ri 209

Description: A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
3bitr3i.1	|- ( ph <-> ps )
3bitr3i.2	|- ( ph <-> ch )
3bitr3i.3	|- ( ps <-> th )

Assertion

Ref	Expression
3bitr3ri	|- ( th <-> ch )

Proof of Theorem 3bitr3ri

Step	Hyp	Ref	Expression
1		3bitr3i.3	. 2 |- ( ps <-> th )
2		3bitr3i.1	. . 3 |- ( ph <-> ps )
3		3bitr3i.2	. . 3 |- ( ph <-> ch )
4	2, 3	bitr3i 184	. 2 |- ( ps <-> ch )
5	1, 4	bitr3i 184	1 |- ( th <-> ch )

Syntax hints:   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  bigolden  896  sb9  1896  sbcco  2836  dfiin2g  3711  dffun6f  4935