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  1885  rspc2  2918  rspc3v  2923  copsexg  4330  chfnrn  5746  ffnfv  5793  f1elima  5897  smoel2  6449  th3q  6787  fiintim  7093  addnnnq0  7636  mulnnnq0  7637  addsrpr  7932  mulsrpr  7933  cau3lem  11625  rescncf  15255