Theorem sylan9 638

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 66	. 2 |- (ph -> (th -> (ps -> ta)))
4	3	imp 418	1 |- ((ph /\ th) -> (ps -> ta))

Syntax hints:   -> wi 4   /\ wa 358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  sbequi  2059  sbal1  2126  rspc2  2961  rspc3v  2965  copsexg  4608  chfnrn  5400  ffnfv  5428  f1elima  5475  isotr  5496