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Mirrors > Home > ILE Home > Th. List > pm2.85dc | GIF version |
Description: Reverse distribution of disjunction over implication, given decidability. Based on theorem *2.85 of [WhiteheadRussell] p. 108. (Contributed by Jim Kingdon, 1-Apr-2018.) |
Ref | Expression |
---|---|
pm2.85dc | ⊢ (DECID φ → (((φ ∨ ψ) → (φ ∨ χ)) → (φ ∨ (ψ → χ)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 742 | . 2 ⊢ (DECID φ ↔ (φ ∨ ¬ φ)) | |
2 | orc 632 | . . . 4 ⊢ (φ → (φ ∨ (ψ → χ))) | |
3 | 2 | a1d 22 | . . 3 ⊢ (φ → (((φ ∨ ψ) → (φ ∨ χ)) → (φ ∨ (ψ → χ)))) |
4 | olc 631 | . . . . . 6 ⊢ (ψ → (φ ∨ ψ)) | |
5 | 4 | imim1i 54 | . . . . 5 ⊢ (((φ ∨ ψ) → (φ ∨ χ)) → (ψ → (φ ∨ χ))) |
6 | orel1 643 | . . . . 5 ⊢ (¬ φ → ((φ ∨ χ) → χ)) | |
7 | 5, 6 | syl9r 67 | . . . 4 ⊢ (¬ φ → (((φ ∨ ψ) → (φ ∨ χ)) → (ψ → χ))) |
8 | olc 631 | . . . 4 ⊢ ((ψ → χ) → (φ ∨ (ψ → χ))) | |
9 | 7, 8 | syl6 29 | . . 3 ⊢ (¬ φ → (((φ ∨ ψ) → (φ ∨ χ)) → (φ ∨ (ψ → χ)))) |
10 | 3, 9 | jaoi 635 | . 2 ⊢ ((φ ∨ ¬ φ) → (((φ ∨ ψ) → (φ ∨ χ)) → (φ ∨ (ψ → χ)))) |
11 | 1, 10 | sylbi 114 | 1 ⊢ (DECID φ → (((φ ∨ ψ) → (φ ∨ χ)) → (φ ∨ (ψ → 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-in2 545 ax-io 629 |
This theorem depends on definitions: df-bi 110 df-dc 742 |
This theorem is referenced by: orimdidc 811 |
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