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Mirrors > Home > MPE Home > Th. List > emptynf | Structured version Visualization version GIF version |
Description: On the empty domain, any variable is effectively nonfree in any formula. (Contributed by Wolf Lammen, 12-Mar-2023.) |
Ref | Expression |
---|---|
emptynf | ⊢ (¬ ∃𝑥⊤ → Ⅎ𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | emptyal 1909 | . 2 ⊢ (¬ ∃𝑥⊤ → ∀𝑥𝜑) | |
2 | nftht 1793 | . 2 ⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑) | |
3 | 1, 2 | syl 17 | 1 ⊢ (¬ ∃𝑥⊤ → Ⅎ𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1535 ⊤wtru 1538 ∃wex 1780 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 |
This theorem depends on definitions: df-bi 209 df-tru 1540 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
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