Theorem bi2anan9r 632

Description: Deduction joining two equivalences to form equivalence of conjunctions.

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 631	. 2 |- ((ph /\ th) -> ((ps /\ ta) <-> (ch /\ et)))
4	3	ancoms 436	1 |- ((th /\ ph) -> ((ps /\ ta) <-> (ch /\ et)))

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

This theorem is referenced by:  axinfnd 4938  ltsopq 5055  ltsosr 5183  lt2msq 5837  metxp 7786  metcnconst 7837

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

Copyright terms: Public domain