Theorem jcn 329

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 161	. 2 ⊢ (𝜑 → ¬ (𝜓 → ¬ ¬ 𝜒))
4		notnotb 307	. . 3 ⊢ (𝜒 ↔ ¬ ¬ 𝜒)
5	4	imbi2i 328	. 2 ⊢ ((𝜓 → 𝜒) ↔ (𝜓 → ¬ ¬ 𝜒))
6	3, 5	sylnibr 321	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 depends on definitions:  df-bi 199

This theorem is referenced by:  isf34lem4  9597  strlem6  29814  hstrlem6  29822  nn0prpw  33198  unblimceq0  33372  relexpmulg  39424  limcrecl  41347  ichnreuop  43008