Theorem jcn 624

Description: Inference joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

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

Assertion

Ref	Expression
jcn	⊢ (𝜑 → ¬ (𝜓 → 𝜒))

Proof of Theorem jcn

Step	Hyp	Ref	Expression
1		jcn.1	. . 3 ⊢ (𝜑 → 𝜓)
2		jcn.2	. . 3 ⊢ (𝜑 → ¬ 𝜒)
3	1, 2	jc 623	. 2 ⊢ (𝜑 → ¬ (𝜓 → ¬ ¬ 𝜒))
4		notnot 601	. . 3 ⊢ (𝜒 → ¬ ¬ 𝜒)
5	4	imim2i 12	. 2 ⊢ ((𝜓 → 𝜒) → (𝜓 → ¬ ¬ 𝜒))
6	3, 5	nsyl 600	1 ⊢ (𝜑 → ¬ (𝜓 → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-in1 586  ax-in2 587

This theorem is referenced by: (None)