Theorem biantr 897

Description: A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117. (Contributed by NM, 18-Aug-1993.)

Assertion

Ref	Expression
biantr	|- (((ph <-> ps) /\ (ch <-> ps)) -> (ph <-> ch))

Proof of Theorem biantr

Step	Hyp	Ref	Expression
1		id 19	. . 3 |- ((ch <-> ps) -> (ch <-> ps))
2	1	bibi2d 309	. 2 |- ((ch <-> ps) -> ((ph <-> ch) <-> (ph <-> ps)))
3	2	biimparc 473	1 |- (((ph <-> ps) /\ (ch <-> ps)) -> (ph <-> ch))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  bm1.1  2338