Theorem pm3.2an3 1171

Description: pm3.2 138 for a triple conjunction. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

Ref	Expression
pm3.2an3	|- ( ph -> ( ps -> ( ch -> ( ph /\ ps /\ ch ) ) ) )

Proof of Theorem pm3.2an3

Step	Hyp	Ref	Expression
1		pm3.2 138	. . 3 |- ( ( ph /\ ps ) -> ( ch -> ( ( ph /\ ps ) /\ ch ) ) )
2	1	ex 114	. 2 |- ( ph -> ( ps -> ( ch -> ( ( ph /\ ps ) /\ ch ) ) ) )
3		df-3an 975	. . 3 |- ( ( ph /\ ps /\ ch ) <-> ( ( ph /\ ps ) /\ ch ) )
4	3	bicomi 131	. 2 |- ( ( ( ph /\ ps ) /\ ch ) <-> ( ph /\ ps /\ ch ) )
5	2, 4	syl8ib 165	1 |- ( ph -> ( ps -> ( ch -> ( ph /\ ps /\ ch ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ 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:  3exp  1197