Theorem nsyli 133

Description: A negated syllogism inference. (Contributed by NM, 3-May-1994.)

Hypotheses

Ref	Expression
nsyli.1	⊢ (φ → (ψ → χ))
nsyli.2	⊢ (θ → ¬ χ)

Assertion

Ref	Expression
nsyli	⊢ (φ → (θ → ¬ ψ))

Proof of Theorem nsyli

Step	Hyp	Ref	Expression
1		nsyli.2	. 2 ⊢ (θ → ¬ χ)
2		nsyli.1	. . 3 ⊢ (φ → (ψ → χ))
3	2	con3d 125	. 2 ⊢ (φ → (¬ χ → ¬ ψ))
4	1, 3	syl5 28	1 ⊢ (φ → (θ → ¬ ψ))

Syntax hints:  ¬ wn 3   → wi 4

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

This theorem is referenced by:  spimehOLD  1821