Theorem biantr 972

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 22	. . 3 |- ( ( ch <-> ps ) -> ( ch <-> ps ) )
2	1	bibi2d 332	. 2 |- ( ( ch <-> ps ) -> ( ( ph <-> ch ) <-> ( ph <-> ps ) ) )
3	2	biimparc 504	1 |- ( ( ( ph <-> ps ) /\ ( ch <-> ps ) ) -> ( ph <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384

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 197  df-an 386

This theorem is referenced by:  bm1.1  2607  bitr3VD  39084  sbcoreleleqVD  39095  trsbcVD  39113  sbcssgVD  39119