![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > hbim | GIF version |
Description: If x is not free in φ and ψ, it is not free in (φ → ψ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.) |
Ref | Expression |
---|---|
hb.1 | ⊢ (φ → ∀xφ) |
hb.2 | ⊢ (ψ → ∀xψ) |
Ref | Expression |
---|---|
hbim | ⊢ ((φ → ψ) → ∀x(φ → ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-4 1397 | . . 3 ⊢ (∀xφ → φ) | |
2 | hb.2 | . . 3 ⊢ (ψ → ∀xψ) | |
3 | 1, 2 | imim12i 53 | . 2 ⊢ ((φ → ψ) → (∀xφ → ∀xψ)) |
4 | ax-i5r 1425 | . 2 ⊢ ((∀xφ → ∀xψ) → ∀x(∀xφ → ψ)) | |
5 | hb.1 | . . . 4 ⊢ (φ → ∀xφ) | |
6 | 5 | imim1i 54 | . . 3 ⊢ ((∀xφ → ψ) → (φ → ψ)) |
7 | 6 | alimi 1341 | . 2 ⊢ (∀x(∀xφ → ψ) → ∀x(φ → ψ)) |
8 | 3, 4, 7 | 3syl 17 | 1 ⊢ ((φ → ψ) → ∀x(φ → ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1240 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-5 1333 ax-gen 1335 ax-4 1397 ax-i5r 1425 |
This theorem is referenced by: hbbi 1437 hbia1 1441 19.21h 1446 19.38 1563 hbsbv 1814 hbmo1 1935 hbmo 1936 moexexdc 1981 2eu4 1990 cleqh 2134 hbral 2347 |
Copyright terms: Public domain | W3C validator |