Theorem 3ancoma 1016

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 266	. . 3 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
2	1	anbi1i 462	. 2 |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ( ps /\ ph ) /\ ch ) )
3		df-3an 1011	. 2 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
4		df-3an 1011	. 2 |- ( ( ps /\ ph /\ ch ) <-> ( ( ps /\ ph ) /\ ch ) )
5	2, 3, 4	3bitr4i 212	1 |- ( ( ph /\ ps /\ ch ) <-> ( ps /\ ph /\ ch ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   /\ w3a 1009

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  3ancomb  1017  3anrev  1019  3anan12  1021  3com12  1238  elfzmlbp  10541  elfzo2  10559  pythagtriplem2  13047  pythagtrip  13064  xpsfrnel  13667