Theorem sylan9 401

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 71	. 2 |- ( ph -> ( th -> ( ps -> ta ) ) )
4	3	imp 122	1 |- ( ( ph /\ th ) -> ( ps -> ta ) )

Syntax hints:   -> wi 4   /\ wa 102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105

This theorem is referenced by:  sbequi  1760  rspc2  2711  rspc3v  2716  trintssmOLD  3892  copsexg  3999  chfnrn  5299  ffnfv  5344  f1elima  5433  smoel2  5941  th3q  6234  addnnnq0  6639  mulnnnq0  6640  addsrpr  6922  mulsrpr  6923  cau3lem  10000