Theorem biantr 814

Description: A transitive law of equivalence. Compare Theorem *4.22 of [WhiteheadRussell] p. 117.

Assertion

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

Proof of Theorem biantr

Step	Hyp	Ref	Expression
1		id 73	. . 3 |- ((ch <-> ps) -> (ch <-> ps))
2	1	bibi2d 680	. 2 |- ((ch <-> ps) -> ((ph <-> ch) <-> (ph <-> ps)))
3	2	biimparc 463	1 |- (((ph <-> ps) /\ (ch <-> ps)) -> (ph <-> ch))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240

This theorem is referenced by:  bitr3 1283  bm1.1 1870  bitr3VD 16673  sbcoreleleqVD 16683  trsbcVD 16701

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-an 242

Copyright terms: Public domain