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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-dfnnf3 | Structured version Visualization version GIF version |
Description: Alternate definition of nonfreeness when sp 2182 is available. (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1785. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-dfnnf3 | ⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nnfea 34064 | . 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) | |
2 | bj-19.21bit 34024 | . . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑 → 𝜑)) | |
3 | bj-19.23bit 34025 | . . 3 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑)) | |
4 | df-bj-nnf 34056 | . . 3 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑))) | |
5 | 2, 3, 4 | sylanbrc 585 | . 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜑) → Ⅎ'𝑥𝜑) |
6 | 1, 5 | impbii 211 | 1 ⊢ (Ⅎ'𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 ∃wex 1780 Ⅎ'wnnf 34055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-bj-nnf 34056 |
This theorem is referenced by: bj-nfnnfTEMP 34087 |
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