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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfea | Structured version Visualization version GIF version |
Description: Nonfreeness implies the equivalent of ax5ea 1914. (Contributed by BJ, 28-Jul-2023.) |
Ref | Expression |
---|---|
bj-nnfea | ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nnfe 34062 | . 2 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) | |
2 | bj-nnfa 34060 | . 2 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) | |
3 | 1, 2 | syld 47 | 1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-bj-nnf 34056 |
This theorem is referenced by: bj-nnfead 34065 bj-nnfnfTEMP 34067 bj-dfnnf3 34086 |
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