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Mirrors > Home > ILE Home > Th. List > pm2.61ddc | GIF version |
Description: Deduction eliminating a decidable antecedent. (Contributed by Jim Kingdon, 4-May-2018.) |
Ref | Expression |
---|---|
pm2.61ddc.1 | ⊢ (φ → (ψ → χ)) |
pm2.61ddc.2 | ⊢ (φ → (¬ ψ → χ)) |
Ref | Expression |
---|---|
pm2.61ddc | ⊢ (DECID ψ → (φ → χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 742 | . 2 ⊢ (DECID ψ ↔ (ψ ∨ ¬ ψ)) | |
2 | pm2.61ddc.1 | . . . 4 ⊢ (φ → (ψ → χ)) | |
3 | 2 | com12 27 | . . 3 ⊢ (ψ → (φ → χ)) |
4 | pm2.61ddc.2 | . . . 4 ⊢ (φ → (¬ ψ → χ)) | |
5 | 4 | com12 27 | . . 3 ⊢ (¬ ψ → (φ → χ)) |
6 | 3, 5 | jaoi 635 | . 2 ⊢ ((ψ ∨ ¬ ψ) → (φ → χ)) |
7 | 1, 6 | sylbi 114 | 1 ⊢ (DECID ψ → (φ → χ)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 628 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: bijadc 775 |
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