Theorem syl8 70

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	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
syl8.2	⊢ (𝜃 → 𝜏)

Assertion

Ref	Expression
syl8	⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))

Proof of Theorem syl8

Step	Hyp	Ref	Expression
1		syl8.1	. 2 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))
2		syl8.2	. . 3 ⊢ (𝜃 → 𝜏)
3	2	a1i 9	. 2 ⊢ (𝜑 → (𝜃 → 𝜏))
4	1, 3	syl6d 69	1 ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))

Syntax hints:   → wi 4

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

This theorem is referenced by:  com45  88  syl8ib  164  imp5a  350  con4biddc  787  3exp  1137  suctr  4176  ssorduni  4231  nneneq  6343  qreccl  8727  bj-inf2vnlem2  10766