Theorem syl8 71

Description: A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)

Hypotheses

Ref	Expression
syl8.1	|- ( ph -> ( ps -> ( ch -> th ) ) )
syl8.2	|- ( th -> ta )

Assertion

Ref	Expression
syl8	|- ( ph -> ( ps -> ( ch -> ta ) ) )

Proof of Theorem syl8

Step	Hyp	Ref	Expression
1		syl8.1	. 2 |- ( ph -> ( ps -> ( ch -> th ) ) )
2		syl8.2	. . 3 |- ( th -> ta )
3	2	a1i 9	. 2 |- ( ph -> ( th -> ta ) )
4	1, 3	syl6d 70	1 |- ( ph -> ( ps -> ( ch -> ta ) ) )

Syntax hints:   -> wi 4

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

This theorem is referenced by:  com45  89  syl8ib  165  imp5a  356  con4biddc  852  3exp  1197  suctr  4406  ssorduni  4471  nneneq  6835  qreccl  9601  bj-inf2vnlem2  14006