Theorem bi2bian9 632

Description: Deduction joining two biconditionals with different antecedents.

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 389	. 2 |- ((ph /\ th) -> (ps <-> ch))
3		bi2an9.2	. . 3 |- (th -> (ta <-> et))
4	3	adantl 388	. 2 |- ((ph /\ th) -> (ta <-> et))
5	2, 4	bibi12d 627	1 |- ((ph /\ th) -> ((ps <-> ta) <-> (ch <-> et)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223

This theorem is referenced by:  uzindOLD 6156

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 147  df-an 225

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