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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nnf-exlim | Structured version Visualization version GIF version |
Description: Proof of the closed form of exlimi 2215 from modalK (compare exlimiv 1931). See also bj-sylget2 34068. (Contributed by BJ, 2-Dec-2023.) |
Ref | Expression |
---|---|
bj-nnf-exlim | ⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exim 1835 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓)) | |
2 | bj-nnfe 34177 | . 2 ⊢ (Ⅎ'𝑥𝜓 → (∃𝑥𝜓 → 𝜓)) | |
3 | 1, 2 | syl9r 78 | 1 ⊢ (Ⅎ'𝑥𝜓 → (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 ∃wex 1781 Ⅎ'wnnf 34170 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-bj-nnf 34171 |
This theorem is referenced by: bj-19.23t 34214 |
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