Theorem syl6an 1434

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  6847  prarloclem5  7498  ltsopr  7594  suplocsrlem  7806  nominpos  9155  ublbneg  9612  absle  11097  rexanre  11228  rexico  11229  climshftlemg  11309  serf0  11359  dvds1lem  11808  dvds2lem  11809  lmconst  13686  addcncntoplem  14021  bj-indind  14654