Theorem 3anbi123d 1206

Description: Deduction joining 3 equivalences to form equivalence of conjunctions. (Contributed by NM, 22-Apr-1994.)

Hypotheses

Ref	Expression
bi3d.1	|- (ph -> (ps <-> ch))
bi3d.2	|- (ph -> (th <-> ta))
bi3d.3	|- (ph -> (et <-> ze))

Assertion

Ref	Expression
3anbi123d	|- (ph -> ((ps /\ th /\ et) <-> (ch /\ ta /\ ze)))

Proof of Theorem 3anbi123d

Step	Hyp	Ref	Expression
1		bi3d.1	. . . 4 |- (ph -> (ps <-> ch))
2		bi3d.2	. . . 4 |- (ph -> (th <-> ta))
3	1, 2	anbi12d 442	. . 3 |- (ph -> ((ps /\ th) <-> (ch /\ ta)))
4		bi3d.3	. . 3 |- (ph -> (et <-> ze))
5	3, 4	anbi12d 442	. 2 |- (ph -> (((ps /\ th) /\ et) <-> ((ch /\ ta) /\ ze)))
6		df-3an 886	. 2 |- ((ps /\ th /\ et) <-> ((ps /\ th) /\ et))
7		df-3an 886	. 2 |- ((ch /\ ta /\ ze) <-> ((ch /\ ta) /\ ze))
8	5, 6, 7	3bitr4g 212	1 |- (ph -> ((ps /\ th /\ et) <-> (ch /\ ta /\ ze)))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 884

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-3an 886

This theorem is referenced by:  3anbi12d  1207  3anbi13d  1208  3anbi23d  1209  sbc3ang  2814  limeq  4080  smoeq  5846  tfrlemi1  5887  ereq1  6049  elinp  6457  iccshftr  8632  iccshftl  8634  iccdil  8636  icccntr  8638  fzaddel  8692  elfzomelpfzo  8857