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  7148  prarloclem5  7868  ltsopr  7964  suplocsrlem  8176  nominpos  9548  ublbneg  10023  wrdsymb0  11353  ccats1pfxeqrex  11503  absle  11872  rexanre  12003  rexico  12004  climshftlemg  12087  serf0  12137  dvds1lem  12588  dvds2lem  12589  lmconst  15408  addcncntoplem  15753  bj-indind  17124