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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfa1 | Structured version Visualization version GIF version | ||
| Description: See nfa1 2150. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-nnfa1 | ⊢ Ⅎ'𝑥∀𝑥𝜑 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hbe1a 2143 | . 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) | |
| 2 | bj-modal4 36716 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | |
| 3 | df-bj-nnf 36726 | . 2 ⊢ (Ⅎ'𝑥∀𝑥𝜑 ↔ ((∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) ∧ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑))) | |
| 4 | 1, 2, 3 | mpbir2an 711 | 1 ⊢ Ⅎ'𝑥∀𝑥𝜑 | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 ∃wex 1778 Ⅎ'wnnf 36725 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-10 2140 ax-12 2176 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-bj-nnf 36726 | 
| This theorem is referenced by: (None) | 
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