Theorem jcnd 164

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 163	. 2 ⊢ (𝜓 → (¬ 𝜒 → ¬ (𝜓 → 𝜒)))
4	1, 2, 3	sylc 66	1 ⊢ (𝜑 → ¬ (𝜓 → 𝜒))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  nf1const  7308  isf34lem4  10372  strlem6  32655  hstrlem6  32663  antnestlaw3lem  36195  antnestlaw1  36196  antnestlaw2  36197  nn0prpw  36867  unblimceq0  37129  relexpmulg  44469  limcrecl  46378  et-sqrtnegnre  47620  ichnreuop  48254