Theorem nsyli 106

Description: A negated syllogism inference.

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.1	. . 3 ⊢ (φ → (ψ → χ))
2	1	con3d 87	. 2 ⊢ (φ → (¬ χ → ¬ ψ))
3		nsyli.2	. 2 ⊢ (θ → ¬ χ)
4	2, 3	syl5 22	1 ⊢ (φ → (θ → ¬ ψ))

Syntax hints:  ¬ wn 1   → wi 2

This theorem is referenced by:  tz7.7 2224  tz7.48-2 2995  nnmord 3189  php 3409  nndomo 3416  isfinite2 3437  eirrv 3449  setind 3492  zornlem3 3605  infxpidmlem10 4942

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

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