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Mirrors > Home > MPE Home > Th. List > spfalw | Structured version Visualization version GIF version |
Description: Version of sp 2041 when 𝜑 is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.) |
Ref | Expression |
---|---|
spfalw.1 | ⊢ ¬ 𝜑 |
Ref | Expression |
---|---|
spfalw | ⊢ (∀𝑥𝜑 → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spfalw.1 | . . 3 ⊢ ¬ 𝜑 | |
2 | 1 | hbth 1720 | . 2 ⊢ (¬ 𝜑 → ∀𝑥 ¬ 𝜑) |
3 | 2 | spnfw 1915 | 1 ⊢ (∀𝑥𝜑 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-6 1875 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: ax6dgen 1992 |
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