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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnfa1 | Structured version Visualization version GIF version |
Description: See nfa1 2149. (Contributed by BJ, 12-Aug-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnfa1 | ⊢ Ⅎ'𝑥∀𝑥𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbe1a 2142 | . 2 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) | |
2 | bj-modal4 36697 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | |
3 | df-bj-nnf 36707 | . 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 1535 ∃wex 1776 Ⅎ'wnnf 36706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-10 2139 ax-12 2175 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-bj-nnf 36707 |
This theorem is referenced by: (None) |
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