Theorem bj-imnimnn 13694

Description: If a formula is implied by both a formula and its negation, then it is not refutable. There is another proof using the inference associated with bj-nnclavius 13693 as its last step. (Contributed by BJ, 27-Oct-2024.)

Hypotheses

Ref	Expression
bj-imnimnn.1	⊢ (𝜑 → 𝜓)
bj-imnimnn.2	⊢ (¬ 𝜑 → 𝜓)

Assertion

Ref	Expression
bj-imnimnn	⊢ ¬ ¬ 𝜓

Proof of Theorem bj-imnimnn

Step	Hyp	Ref	Expression
1		bj-imnimnn.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	con3i 627	. 2 ⊢ (¬ 𝜓 → ¬ 𝜑)
3		bj-imnimnn.2	. . 3 ⊢ (¬ 𝜑 → 𝜓)
4	3	con3i 627	. 2 ⊢ (¬ 𝜓 → ¬ ¬ 𝜑)
5	2, 4	pm2.65i 634	1 ⊢ ¬ ¬ 𝜓

Syntax hints:  ¬ wn 3   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 609  ax-in2 610

This theorem is referenced by:  bj-nnst  13699