Theorem bi2anan9r 695

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 694	. 2 |- ((ph /\ th) -> ((ps /\ ta) <-> (ch /\ et)))
4	3	ancoms 484	1 |- ((th /\ ph) -> ((ps /\ ta) <-> (ch /\ et)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240

This theorem is referenced by:  axinfnd 6110  ltsopq 6227  ltsosr 6355  lt2msqi 7064  metxp 9111  metcnconst 9163

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 164  df-an 242

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