Theorem bi2anan9r 918

Description: Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)

Hypotheses

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

Assertion

Ref	Expression
bi2anan9r	|- ( ( th /\ ph ) -> ( ( ps /\ ta ) <-> ( ch /\ et ) ) )

Proof of Theorem bi2anan9r

Step	Hyp	Ref	Expression
1		bi2an9.1	. . 3 |- ( ph -> ( ps <-> ch ) )
2		bi2an9.2	. . 3 |- ( th -> ( ta <-> et ) )
3	1, 2	bi2anan9 917	. 2 |- ( ( ph /\ th ) -> ( ( ps /\ ta ) <-> ( ch /\ et ) ) )
4	3	ancoms 469	1 |- ( ( th /\ ph ) -> ( ( 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:  efrn2lp  5096  ltsosr  9915  seqf1olem2  12841  seqf1o  12842  pcval  15549  uspgr2wlkeq  26542  fneval  32347  prtlem5  34145  rmydioph  37581  wepwsolem  37612  aomclem8  37631  sprsymrelfolem2  41743