Theorem syl6an 1479

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 ) )

Syntax hints:   -> wi 4   /\ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia3 108

This theorem is referenced by:  mapxpen  7114  prarloclem5  7831  ltsopr  7927  suplocsrlem  8139  nominpos  9493  ublbneg  9963  wrdsymb0  11282  ccats1pfxeqrex  11432  absle  11799  rexanre  11930  rexico  11931  climshftlemg  12012  serf0  12062  dvds1lem  12513  dvds2lem  12514  lmconst  15207  addcncntoplem  15552  bj-indind  16828