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Mirrors > Home > MPE Home > Th. List > hbth | Structured version Visualization version GIF version |
Description: No variable is
(effectively) free in a theorem.
This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form ⊢ (𝜑 → ∀𝑥𝜑) from smaller formulas of this form. These are useful for constructing hypotheses that state "𝑥 is (effectively) not free in 𝜑". (Contributed by NM, 11-May-1993.) This hb* idiom is generally being replaced by the nf* idiom (see nfth 1783), but keeps its interest in some cases. (Revised by BJ, 23-Sep-2022.) |
Ref | Expression |
---|---|
hbth.1 | ⊢ 𝜑 |
Ref | Expression |
---|---|
hbth | ⊢ (𝜑 → ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbth.1 | . . 3 ⊢ 𝜑 | |
2 | 1 | ax-gen 1777 | . 2 ⊢ ∀𝑥𝜑 |
3 | 2 | a1i 11 | 1 ⊢ (𝜑 → ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-gen 1777 |
This theorem is referenced by: speiv 1953 spfalw 1981 |
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