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Mirrors > Home > MPE Home > Th. List > emptyal | Structured version Visualization version GIF version |
Description: On the empty domain, any universally quantified formula is true. (Contributed by Wolf Lammen, 12-Mar-2023.) |
Ref | Expression |
---|---|
emptyal | ⊢ (¬ ∃𝑥⊤ → ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | emptyex 1907 | . 2 ⊢ (¬ ∃𝑥⊤ → ¬ ∃𝑥 ¬ 𝜑) | |
2 | alex 1825 | . 2 ⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑) | |
3 | 1, 2 | sylibr 236 | 1 ⊢ (¬ ∃𝑥⊤ → ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1534 ⊤wtru 1537 ∃wex 1779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
This theorem depends on definitions: df-bi 209 df-tru 1539 df-ex 1780 |
This theorem is referenced by: emptynf 1909 |
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