Theorem 3anbi123i 1183

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 457	. . 3 |- ( ( ph /\ ch ) <-> ( ps /\ th ) )
4		bi3.3	. . 3 |- ( ta <-> et )
5	3, 4	anbi12i 457	. 2 |- ( ( ( ph /\ ch ) /\ ta ) <-> ( ( ps /\ th ) /\ et ) )
6		df-3an 975	. 2 |- ( ( ph /\ ch /\ ta ) <-> ( ( ph /\ ch ) /\ ta ) )
7		df-3an 975	. 2 |- ( ( ps /\ th /\ et ) <-> ( ( ps /\ th ) /\ et ) )
8	5, 6, 7	3bitr4i 211	1 |- ( ( ph /\ ch /\ ta ) <-> ( ps /\ th /\ et ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 973

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-3an 975

This theorem is referenced by:  3anbi1i  1185  3anbi2i  1186  3anbi3i  1187  syl3anb  1276  ne3anior  2428  isstructim  12430