Theorem 3anbi123d 1207

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 887	. 2 |- ( ( ps /\ th /\ et ) <-> ( ( ps /\ th ) /\ et ) )
7		df-3an 887	. 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 885

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 887

This theorem is referenced by:  3anbi12d  1208  3anbi13d  1209  3anbi23d  1210  sbc3ang  2820  limeq  4114  smoeq  5905  tfrlemi1  5946  ereq1  6113  elinp  6572  iccshftr  8862  iccshftl  8864  iccdil  8866  icccntr  8868  fzaddel  8922  elfzomelpfzo  9087