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Mirrors > Home > ILE Home > Th. List > pm2.54dc | GIF version |
Description: Deriving disjunction from implication for a decidable proposition. Based on theorem *2.54 of [WhiteheadRussell] p. 107. The converse, pm2.53 641, holds whether the proposition is decidable or not. (Contributed by Jim Kingdon, 26-Mar-2018.) |
Ref | Expression |
---|---|
pm2.54dc | ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dcn 746 | . 2 ⊢ (DECID 𝜑 → DECID ¬ 𝜑) | |
2 | notnotrdc 751 | . . . . 5 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → 𝜑)) | |
3 | orc 633 | . . . . 5 ⊢ (𝜑 → (𝜑 ∨ 𝜓)) | |
4 | 2, 3 | syl6 29 | . . . 4 ⊢ (DECID 𝜑 → (¬ ¬ 𝜑 → (𝜑 ∨ 𝜓))) |
5 | 4 | a1d 22 | . . 3 ⊢ (DECID 𝜑 → (DECID ¬ 𝜑 → (¬ ¬ 𝜑 → (𝜑 ∨ 𝜓)))) |
6 | olc 632 | . . . 4 ⊢ (𝜓 → (𝜑 ∨ 𝜓)) | |
7 | 6 | a1i 9 | . . 3 ⊢ (DECID 𝜑 → (𝜓 → (𝜑 ∨ 𝜓))) |
8 | 5, 7 | jaddc 761 | . 2 ⊢ (DECID 𝜑 → (DECID ¬ 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓)))) |
9 | 1, 8 | mpd 13 | 1 ⊢ (DECID 𝜑 → ((¬ 𝜑 → 𝜓) → (𝜑 ∨ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 629 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-in1 544 ax-in2 545 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 |
This theorem is referenced by: dfordc 791 pm2.68dc 793 pm4.79dc 809 pm5.11dc 815 |
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