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  2852  rspc3v  2857  copsexg  4244  chfnrn  5627  ffnfv  5674  f1elima  5773  smoel2  6303  th3q  6639  fiintim  6927  addnnnq0  7447  mulnnnq0  7448  addsrpr  7743  mulsrpr  7744  cau3lem  11122  rescncf  14038