Theorem 3bitr2ri 265

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 243	. 2 |- (ph <-> ch)
4		3bitr2i.3	. 2 |- (ch <-> th)
5	3, 4	bitr2i 241	1 |- (th <-> ph)

This proof depends on syntax axioms:   <-> wb 176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177

This theorem is used by:  sbnf2  2108  ssrab  3345  unipw1  4326  dmopab3  4918  dfres2  5003  ssrnres  5060  df2nd2  5112  dfnnc3  5886  enex  6032