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Mirrors > Home > ILE Home > Th. List > jaddc | GIF version |
Description: Deduction forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 26-Mar-2018.) |
Ref | Expression |
---|---|
jaddc.1 | ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝜃))) |
jaddc.2 | ⊢ (𝜑 → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
jaddc | ⊢ (𝜑 → (DECID 𝜓 → ((𝜓 → 𝜒) → 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jaddc.2 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜃)) | |
2 | 1 | imim2d 48 | . 2 ⊢ (𝜑 → ((𝜓 → 𝜒) → (𝜓 → 𝜃))) |
3 | jaddc.1 | . . 3 ⊢ (𝜑 → (DECID 𝜓 → (¬ 𝜓 → 𝜃))) | |
4 | pm2.6dc 759 | . . 3 ⊢ (DECID 𝜓 → ((¬ 𝜓 → 𝜃) → ((𝜓 → 𝜃) → 𝜃))) | |
5 | 3, 4 | sylcom 25 | . 2 ⊢ (𝜑 → (DECID 𝜓 → ((𝜓 → 𝜃) → 𝜃))) |
6 | 2, 5 | syl5d 62 | 1 ⊢ (𝜑 → (DECID 𝜓 → ((𝜓 → 𝜒) → 𝜃))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 DECID wdc 742 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 |
This theorem is referenced by: pm2.54dc 790 |
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