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Mirrors > Home > ILE Home > Th. List > jcn | GIF version |
Description: Inference joining the consequents of two premises. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
jcn.1 | ⊢ (𝜑 → 𝜓) |
jcn.2 | ⊢ (𝜑 → ¬ 𝜒) |
Ref | Expression |
---|---|
jcn | ⊢ (𝜑 → ¬ (𝜓 → 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jcn.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
2 | jcn.2 | . . 3 ⊢ (𝜑 → ¬ 𝜒) | |
3 | 1, 2 | jc 623 | . 2 ⊢ (𝜑 → ¬ (𝜓 → ¬ ¬ 𝜒)) |
4 | notnot 601 | . . 3 ⊢ (𝜒 → ¬ ¬ 𝜒) | |
5 | 4 | imim2i 12 | . 2 ⊢ ((𝜓 → 𝜒) → (𝜓 → ¬ ¬ 𝜒)) |
6 | 3, 5 | nsyl 600 | 1 ⊢ (𝜑 → ¬ (𝜓 → 𝜒)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-in1 586 ax-in2 587 |
This theorem is referenced by: (None) |
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