Theorem syl6an 1483

Description: A syllogism deduction combined with conjoining antecedents. (Contributed by Alan Sare, 28-Oct-2011.)

Hypotheses

Ref	Expression
syl6an.1	|- ( ph -> ps )
syl6an.2	|- ( ph -> ( ch -> th ) )
syl6an.3	|- ( ( ps /\ th ) -> ta )

Assertion

Ref	Expression
syl6an	|- ( ph -> ( ch -> ta ) )

Proof of Theorem syl6an

Step	Hyp	Ref	Expression
1		syl6an.2	. . 3 |- ( ph -> ( ch -> th ) )
2		syl6an.1	. . 3 |- ( ph -> ps )
3	1, 2	jctild 316	. 2 |- ( ph -> ( ch -> ( ps /\ th ) ) )
4		syl6an.3	. 2 |- ( ( ps /\ th ) -> ta )
5	3, 4	syl6 33	1 |- ( ph -> ( ch -> ta ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104

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

This theorem is used by:  mapxpen  7152  prarloclem5  7873  ltsopr  7969  suplocsrlem  8181  nominpos  9554  ublbneg  10029  wrdsymb0  11359  ccats1pfxeqrex  11509  absle  11878  rexanre  12009  rexico  12010  climshftlemg  12093  serf0  12143  dvds1lem  12594  dvds2lem  12595  lmconst  15442  addcncntoplem  15787  bj-indind  17163