Theorem 3ancoma 975

Description: Commutation law for triple conjunction. (Contributed by NM, 21-Apr-1994.)

Assertion

Ref	Expression
3ancoma	|- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ph /\ ch ) )

Proof of Theorem 3ancoma

Step	Hyp	Ref	Expression
1		ancom 264	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
2	1	anbi1i 454	. 2 |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ps /\ ph ) /\ ch ) )
3		df-3an 970	. 2 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
4		df-3an 970	. 2 |- ( ( ps /\ ph /\ ch ) <-> ( ( ps /\ ph ) /\ ch ) )
5	2, 3, 4	3bitr4i 211	1 |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ph /\ ch ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 968

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 970

This theorem is referenced by:  3ancomb  976  3anrev  978  3anan12  980  3com12  1197  elfzmlbp  10063  elfzo2  10081  pythagtriplem2  12194  pythagtrip  12211