Theorem syl6an 1476

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  7029  prarloclem5  7710  ltsopr  7806  suplocsrlem  8018  nominpos  9372  ublbneg  9837  wrdsymb0  11136  ccats1pfxeqrex  11286  absle  11640  rexanre  11771  rexico  11772  climshftlemg  11853  serf0  11903  dvds1lem  12353  dvds2lem  12354  lmconst  14930  addcncntoplem  15275  bj-indind  16463