Theorem bi2bian9 845

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 451	. 2 |- ((ph /\ th) -> (ps <-> ch))
3		bi2an9.2	. . 3 |- (th -> (ta <-> et))
4	3	adantl 452	. 2 |- ((ph /\ th) -> (ta <-> et))
5	2, 4	bibi12d 312	1 |- ((ph /\ th) -> ((ps <-> ta) <-> (ch <-> et)))

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: (None)