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Mirrors > Home > ILE Home > Th. List > 19.32dc | GIF version |
Description: Theorem 19.32 of [Margaris] p. 90, where 𝜑 is decidable. (Contributed by Jim Kingdon, 4-Jun-2018.) |
Ref | Expression |
---|---|
19.32dc.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
19.32dc | ⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 19.32dc.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nfn 1548 | . . . 4 ⊢ Ⅎ𝑥 ¬ 𝜑 |
3 | 2 | 19.21 1475 | . . 3 ⊢ (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓)) |
4 | 3 | a1i 9 | . 2 ⊢ (DECID 𝜑 → (∀𝑥(¬ 𝜑 → 𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))) |
5 | 1 | nfdc 1549 | . . 3 ⊢ Ⅎ𝑥DECID 𝜑 |
6 | dfordc 791 | . . 3 ⊢ (DECID 𝜑 → ((𝜑 ∨ 𝜓) ↔ (¬ 𝜑 → 𝜓))) | |
7 | 5, 6 | albid 1506 | . 2 ⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ ∀𝑥(¬ 𝜑 → 𝜓))) |
8 | dfordc 791 | . 2 ⊢ (DECID 𝜑 → ((𝜑 ∨ ∀𝑥𝜓) ↔ (¬ 𝜑 → ∀𝑥𝜓))) | |
9 | 4, 7, 8 | 3bitr4d 209 | 1 ⊢ (DECID 𝜑 → (∀𝑥(𝜑 ∨ 𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ∨ wo 629 DECID wdc 742 ∀wal 1241 Ⅎwnf 1349 |
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 ax-5 1336 ax-gen 1338 ax-ie2 1383 ax-4 1400 ax-ial 1427 ax-i5r 1428 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-tru 1246 df-fal 1249 df-nf 1350 |
This theorem is referenced by: (None) |
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