Theorem biantr 859

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 221	. 2 |- ( ( ch <-> ps ) -> ( ( ph <-> ch ) <-> ( ph <-> ps ) ) )
3	2	biimparc 283	1 |- ( ( ( ph <-> ps ) /\ ( ch <-> ps ) ) -> ( ph <-> ch ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110

This theorem is referenced by:  bm1.1  2025