Theorem sylan9r 414

Description: Nested syllogism inference conjoining dissimilar antecedents. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
sylan9r.1	|- ( ph -> ( ps -> ch ) )
sylan9r.2	|- ( th -> ( ch -> ta ) )

Assertion

Ref	Expression
sylan9r	|- ( ( th /\ ph ) -> ( ps -> ta ) )

Proof of Theorem sylan9r

Step	Hyp	Ref	Expression
1		sylan9r.1	. . 3 |- ( ph -> ( ps -> ch ) )
2		sylan9r.2	. . 3 |- ( th -> ( ch -> ta ) )
3	1, 2	syl9r 73	. 2 |- ( th -> ( ph -> ( ps -> ta ) ) )
4	3	imp 124	1 |- ( ( th /\ ph ) -> ( ps -> 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-ia1 106  ax-ia2 107

This theorem is used by:  spimt  1789  sbequi  1892  updjudhf  7419  genpcdl  7886  genpcuu  7887  iccsupr  10368  reuccatpfxs1lem  11518  climuni  12059  tgcn  15309  metrest  15607