Theorem bi2bian9 919

Description: Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)

Hypotheses

Ref	Expression
bi2an9.1	|- ( ph -> ( ps <-> ch ) )
bi2an9.2	|- ( th -> ( ta <-> et ) )

Assertion

Ref	Expression
bi2bian9	|- ( ( ph /\ th ) -> ( ( ps <-> ta ) <-> ( ch <-> et ) ) )

Proof of Theorem bi2bian9

Step	Hyp	Ref	Expression
1		bi2an9.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2	1	adantr 481	. 2 |- ( ( ph /\ th ) -> ( ps <-> ch ) )
3		bi2an9.2	. . 3 |- ( th -> ( ta <-> et ) )
4	3	adantl 482	. 2 |- ( ( ph /\ th ) -> ( ta <-> et ) )
5	2, 4	bibi12d 335	1 |- ( ( ph /\ th ) -> ( ( ps <-> ta ) <-> ( ch <-> et ) ) )

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:  releccnveq  34022  wepwsolem  37612  aomclem8  37631