Theorem bi3an 606

Description: Join 3 biconditionals with conjunction.

Hypotheses

Ref	Expression
bi3.1	|- (ph <-> ps)
bi3.2	|- (ch <-> th)
bi3.3	|- (ta <-> et)

Assertion

Ref	Expression
bi3an	|- ((ph /\ ch /\ ta ) <-> (ps /\ th /\ et))

Proof of Theorem bi3an

Step	Hyp	Ref	Expression
1		bi3.1	. . . 4 |- (ph <-> ps)
2		bi3.2	. . . 4 |- (ch <-> th)
3	1, 2	anbi12i 369	. . 3 |- ((ph /\ ch) <-> (ps /\ th))
4		bi3.3	. . 3 |- (ta <-> et)
5	3, 4	anbi12i 369	. 2 |- (((ph /\ ch) /\ ta ) <-> ((ps /\ th) /\ et))
6		df-3an 583	. 2 |- ((ph /\ ch /\ ta ) <-> ((ph /\ ch) /\ ta ))
7		df-3an 583	. 2 |- ((ps /\ th /\ et) <-> ((ps /\ th) /\ et))
8	5, 6, 7	3bitr4 158	1 |- ((ph /\ ch /\ ta ) <-> (ps /\ th /\ et))

Syntax hints:   <-> wb 127   /\ wa 196   /\ w3a 581

This theorem is referenced by:  epne3 2182

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583

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