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Mirrors > Home > ILE Home > Th. List > jadc | GIF version |
Description: Inference forming an implication from the antecedents of two premises, where a decidable antecedent is negated. (Contributed by Jim Kingdon, 25-Mar-2018.) |
Ref | Expression |
---|---|
jadc.1 | ⊢ (DECID φ → (¬ φ → χ)) |
jadc.2 | ⊢ (ψ → χ) |
Ref | Expression |
---|---|
jadc | ⊢ (DECID φ → ((φ → ψ) → χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | jadc.2 | . . 3 ⊢ (ψ → χ) | |
2 | 1 | imim2i 12 | . 2 ⊢ ((φ → ψ) → (φ → χ)) |
3 | jadc.1 | . . 3 ⊢ (DECID φ → (¬ φ → χ)) | |
4 | pm2.6dc 758 | . . 3 ⊢ (DECID φ → ((¬ φ → χ) → ((φ → χ) → χ))) | |
5 | 3, 4 | mpd 13 | . 2 ⊢ (DECID φ → ((φ → χ) → χ)) |
6 | 2, 5 | syl5 28 | 1 ⊢ (DECID φ → ((φ → ψ) → χ)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 DECID wdc 741 |
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 629 |
This theorem depends on definitions: df-bi 110 df-dc 742 |
This theorem is referenced by: pm5.71dc 867 |
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