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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfa | Structured version Visualization version GIF version | ||
| Description: Nonfreeness implies the equivalent of ax-5 1910. See nf5r 2194. (Contributed by BJ, 28-Jul-2023.) |
| Ref | Expression |
|---|---|
| bj-nnfa | ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-bj-nnf 36725 | . 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑))) | |
| 2 | 1 | simprbi 496 | 1 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 ∃wex 1779 Ⅎ'wnnf 36724 |
| 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 207 df-an 396 df-bj-nnf 36725 |
| This theorem is referenced by: bj-nnfad 36730 bj-nnfai 36731 bj-nnfea 36735 bj-nnfim1 36745 bj-nnfim2 36746 bj-nnf-alrim 36756 bj-19.23t 36771 bj-19.37im 36773 bj-19.42t 36774 bj-sbft 36776 |
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