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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 1545 | . . . 4 ⊢ Ⅎx ¬ φ |
3 | 2 | 19.21 1472 | . . 3 ⊢ (∀x(¬ φ → ψ) ↔ (¬ φ → ∀xψ)) |
4 | 3 | a1i 9 | . 2 ⊢ (DECID φ → (∀x(¬ φ → ψ) ↔ (¬ φ → ∀xψ))) |
5 | 1 | nfdc 1546 | . . 3 ⊢ ℲxDECID φ |
6 | dfordc 790 | . . 3 ⊢ (DECID φ → ((φ ∨ ψ) ↔ (¬ φ → ψ))) | |
7 | 5, 6 | albid 1503 | . 2 ⊢ (DECID φ → (∀x(φ ∨ ψ) ↔ ∀x(¬ φ → ψ))) |
8 | dfordc 790 | . 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 628 DECID wdc 741 ∀wal 1240 Ⅎwnf 1346 |
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 629 ax-5 1333 ax-gen 1335 ax-ie2 1380 ax-4 1397 ax-ial 1424 ax-i5r 1425 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-tru 1245 df-fal 1248 df-nf 1347 |
This theorem is referenced by: (None) |
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