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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 1913. See nf5r 2187. (Contributed by BJ, 28-Jul-2023.) |
Ref | Expression |
---|---|
bj-nnfa | ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-nnf 35121 | . 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑))) | |
2 | 1 | simprbi 497 | 1 ⊢ (Ⅎ'𝑥𝜑 → (𝜑 → ∀𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 ∃wex 1781 Ⅎ'wnnf 35120 |
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 206 df-an 397 df-bj-nnf 35121 |
This theorem is referenced by: bj-nnfad 35126 bj-nnfai 35127 bj-nnfea 35131 bj-nnfim1 35141 bj-nnfim2 35142 bj-nnf-alrim 35152 bj-19.23t 35167 bj-19.37im 35169 bj-19.42t 35170 bj-sbft 35172 |
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