Theorem syl7 69

Description: A syllogism rule of inference. The second premise is used to replace the third antecedent of the first premise. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Aug-2012.)

Hypotheses

Ref	Expression
syl7.1	|- ( ph -> ps )
syl7.2	|- ( ch -> ( th -> ( ps -> ta ) ) )

Assertion

Ref	Expression
syl7	|- ( ch -> ( th -> ( ph -> ta ) ) )

Proof of Theorem syl7

Step	Hyp	Ref	Expression
1		syl7.1	. . 3 |- ( ph -> ps )
2	1	a1i 9	. 2 |- ( ch -> ( ph -> ps ) )
3		syl7.2	. 2 |- ( ch -> ( th -> ( ps -> ta ) ) )
4	2, 3	syl5d 68	1 |- ( ch -> ( th -> ( ph -> 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:  syl7bi  165  syl3an3  1273  fvmptt  5610  nneneq  6860  pr2nelem  7193  ndvdssub  11938