Theorem sylan9 409

Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.)

Hypotheses

Ref	Expression
sylan9.1	|- ( ph -> ( ps -> ch ) )
sylan9.2	|- ( th -> ( ch -> ta ) )

Assertion

Ref	Expression
sylan9	|- ( ( ph /\ th ) -> ( ps -> ta ) )

Proof of Theorem sylan9

Step	Hyp	Ref	Expression
1		sylan9.1	. . 3 |- ( ph -> ( ps -> ch ) )
2		sylan9.2	. . 3 |- ( th -> ( ch -> ta ) )
3	1, 2	syl9 72	. 2 |- ( ph -> ( th -> ( ps -> ta ) ) )
4	3	imp 124	1 |- ( ( ph /\ th ) -> ( ps -> ta ) )

Syntax hints:   -> wi 4   /\ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107

This theorem is referenced by:  sbequi  1839  rspc2  2853  rspc3v  2858  copsexg  4245  chfnrn  5628  ffnfv  5675  f1elima  5774  smoel2  6304  th3q  6640  fiintim  6928  addnnnq0  7448  mulnnnq0  7449  addsrpr  7744  mulsrpr  7745  cau3lem  11123  rescncf  14071