Theorem jcnd 165

Description: Deduction joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Proof shortened by Wolf Lammen, 10-Apr-2024.)

Hypotheses

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

Assertion

Ref	Expression
jcnd	⊢ (𝜑 → ¬ (𝜓 → 𝜒))

Proof of Theorem jcnd

Step	Hyp	Ref	Expression
1		jcnd.1	. 2 ⊢ (𝜑 → 𝜓)
2		jcnd.2	. 2 ⊢ (𝜑 → ¬ 𝜒)
3		jcn 164	. 2 ⊢ (𝜓 → (¬ 𝜒 → ¬ (𝜓 → 𝜒)))
4	1, 2, 3	sylc 65	1 ⊢ (𝜑 → ¬ (𝜓 → 𝜒))

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:  norassOLD  1533  nf1const  7056  isf34lem4  9796  strlem6  30031  hstrlem6  30039  nn0prpw  33695  unblimceq0  33870  relexpmulg  40129  limcrecl  41984  ichnreuop  43708