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Mirrors > Home > MPE Home > Th. List > spvw | Structured version Visualization version GIF version |
Description: Version of sp 2176 when 𝑥 does not occur in 𝜑. Converse of ax-5 1913. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) Shorten 19.3v 1985. (Revised by Wolf Lammen, 20-Oct-2023.) |
Ref | Expression |
---|---|
spvw | ⊢ (∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1913 | . 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) | |
2 | 1 | spnfw 1983 | 1 ⊢ (∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 |
This theorem depends on definitions: df-bi 206 df-ex 1783 |
This theorem is referenced by: 19.3v 1985 |
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