Theorem 3bitr2ri 209

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
3bitr2ri	|- ( th <-> ph )

Proof of Theorem 3bitr2ri

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	bitr2i 185	1 |- ( th <-> ph )

This proof depends on syntax axioms:   <-> wb 105

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117

This theorem is used by:  sbnf2  2041  ssrab  3326  rabn0m  3549  unidif0  4304  relop  4930  dmopab3  4994  restidsing  5119  issref  5170  fununi  5449  cnvoprab  6470  ssfirab  7244  rabid1o  17034