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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfnfTEMP | Structured version Visualization version GIF version |
Description: New nonfreeness implies old nonfreeness on implicational calculus (the proof indicates it uses ax-3 8 because of set.mm's definition of the biconditional, but the proof actually holds in minimal implicational calculus). (Contributed by BJ, 28-Jul-2023.) The proof should not rely on df-nf 1785 except via df-nf 1785 directly. (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnfnfTEMP | ⊢ (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nnfea 34071 | . 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) | |
2 | df-nf 1785 | . 2 ⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑)) | |
3 | 1, 2 | sylibr 236 | 1 ⊢ (Ⅎ'𝑥𝜑 → Ⅎ𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 ∃wex 1780 Ⅎwnf 1784 Ⅎ'wnnf 34062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 209 df-an 399 df-nf 1785 df-bj-nnf 34063 |
This theorem is referenced by: (None) |
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