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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-imnimnn | GIF version |
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 13737 as its last step. (Contributed by BJ, 27-Oct-2024.) |
Ref | Expression |
---|---|
bj-imnimnn.1 | ⊢ (𝜑 → 𝜓) |
bj-imnimnn.2 | ⊢ (¬ 𝜑 → 𝜓) |
Ref | Expression |
---|---|
bj-imnimnn | ⊢ ¬ ¬ 𝜓 |
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 |
Colors of variables: wff set class |
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 13743 |
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