Theorem 3anbi123i 821

Description: Join 3 biconditionals with conjunction.

Hypotheses

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

Assertion

Ref	Expression
3anbi123i	|- ((ph /\ ch /\ ta) <-> (ps /\ th /\ et))

Proof of Theorem 3anbi123i

Step	Hyp	Ref	Expression
1		bi3.1	. . . 4 |- (ph <-> ps)
2		bi3.2	. . . 4 |- (ch <-> th)
3	1, 2	anbi12i 482	. . 3 |- ((ph /\ ch) <-> (ps /\ th))
4		bi3.3	. . 3 |- (ta <-> et)
5	3, 4	anbi12i 482	. 2 |- (((ph /\ ch) /\ ta) <-> ((ps /\ th) /\ et))
6		df-3an 776	. 2 |- ((ph /\ ch /\ ta) <-> ((ph /\ ch) /\ ta))
7		df-3an 776	. 2 |- ((ps /\ th /\ et) <-> ((ps /\ th) /\ et))
8	5, 6, 7	3bitr4 183	1 |- ((ph /\ ch /\ ta) <-> (ps /\ th /\ et))

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 774

This theorem is referenced by:  3anbi1i 823  3anbi2i 824  3anbi3i 825  syl3anb 868  epne3 2925  elfz2nn0t 6435  sspval 8329  efifolem2 8657  ishoma 10595

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  df-3an 776

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