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Mirrors > Home > NFE Home > Th. List > hbth | 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, 5-Aug-1993.) |
Ref | Expression |
---|---|
hbth.1 | ⊢ φ |
Ref | Expression |
---|---|
hbth | ⊢ (φ → ∀xφ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbth.1 | . . 3 ⊢ φ | |
2 | 1 | ax-gen 1546 | . 2 ⊢ ∀xφ |
3 | 2 | a1i 10 | 1 ⊢ (φ → ∀xφ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 |
This theorem was proved from axioms: ax-1 5 ax-mp 8 ax-gen 1546 |
This theorem is referenced by: nfth 1553 spfalw 1672 spimehOLD 1821 |
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