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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfe | Structured version Visualization version GIF version |
Description: Nonfreeness implies the equivalent of ax5e 1907. (Contributed by BJ, 28-Jul-2023.) |
Ref | Expression |
---|---|
bj-nnfe | ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bj-nnf 36109 | . 2 ⊢ (Ⅎ'𝑥𝜑 ↔ ((∃𝑥𝜑 → 𝜑) ∧ (𝜑 → ∀𝑥𝜑))) | |
2 | 1 | simplbi 497 | 1 ⊢ (Ⅎ'𝑥𝜑 → (∃𝑥𝜑 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∃wex 1773 Ⅎ'wnnf 36108 |
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 396 df-bj-nnf 36109 |
This theorem is referenced by: bj-nnfed 36117 bj-nnfei 36118 bj-nnfea 36119 bj-nnfim1 36129 bj-nnfim2 36130 bj-nnf-exlim 36141 bj-19.21t 36154 bj-19.36im 36156 bj-19.42t 36158 bj-sbft 36160 |
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