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Mirrors > Home > ILE Home > Th. List > pm5.55dc | GIF version |
Description: A disjunction is equivalent to one of its disjuncts, given a decidable disjunct. Based on theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.) |
Ref | Expression |
---|---|
pm5.55dc | ⊢ (DECID 𝜑 → (((𝜑 ∨ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∨ 𝜓) ↔ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dc 743 | . 2 ⊢ (DECID 𝜑 ↔ (𝜑 ∨ ¬ 𝜑)) | |
2 | biort 738 | . . . 4 ⊢ (𝜑 → (𝜑 ↔ (𝜑 ∨ 𝜓))) | |
3 | 2 | bicomd 129 | . . 3 ⊢ (𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜑)) |
4 | biorf 663 | . . . 4 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | |
5 | 4 | bicomd 129 | . . 3 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
6 | 3, 5 | orim12i 676 | . 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → (((𝜑 ∨ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∨ 𝜓) ↔ 𝜓))) |
7 | 1, 6 | sylbi 114 | 1 ⊢ (DECID 𝜑 → (((𝜑 ∨ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∨ 𝜓) ↔ 𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ∨ 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-in2 545 ax-io 630 |
This theorem depends on definitions: df-bi 110 df-dc 743 |
This theorem is referenced by: (None) |
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