Theorem 3anbi123i 1140

Description: Join 3 biconditionals with conjunction. (Contributed by NM, 21-Apr-1994.)

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 678	. . 3 |- ((ph /\ ch) <-> (ps /\ th))
4		bi3.3	. . 3 |- (ta <-> et)
5	3, 4	anbi12i 678	. 2 |- (((ph /\ ch) /\ ta) <-> ((ps /\ th) /\ et))
6		df-3an 936	. 2 |- ((ph /\ ch /\ ta) <-> ((ph /\ ch) /\ ta))
7		df-3an 936	. 2 |- ((ps /\ th /\ et) <-> ((ps /\ th) /\ et))
8	5, 6, 7	3bitr4i 268	1 |- ((ph /\ ch /\ ta) <-> (ps /\ th /\ et))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934

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 177  df-an 360  df-3an 936

This theorem is referenced by:  3anbi1i  1142  3anbi2i  1143  3anbi3i  1144  syl3anb  1225  cadnot  1394  opksnelsik  4266  eloprabga  5579  restxp  5787  oqelins4  5795  xpassen  6058  mucass  6136  taddc  6230  letc  6232