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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 1908. See nf5r 2192. (Contributed by BJ, 28-Jul-2023.) |
Ref | Expression |
---|---|
bj-nnfa | ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-nnf 36707 | . 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 1535 ∃wex 1776 Ⅎ'wnnf 36706 |
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 36707 |
This theorem is referenced by: bj-nnfad 36712 bj-nnfai 36713 bj-nnfea 36717 bj-nnfim1 36727 bj-nnfim2 36728 bj-nnf-alrim 36738 bj-19.23t 36753 bj-19.37im 36755 bj-19.42t 36756 bj-sbft 36758 |
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