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Mirrors > Home > NFE Home > Th. List > nfimd | GIF version |
Description: If in a context x is not free in ψ and χ, it is not free in (ψ → χ). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) |
Ref | Expression |
---|---|
nfimd.1 | ⊢ (φ → Ⅎxψ) |
nfimd.2 | ⊢ (φ → Ⅎxχ) |
Ref | Expression |
---|---|
nfimd | ⊢ (φ → Ⅎx(ψ → χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfimd.1 | . 2 ⊢ (φ → Ⅎxψ) | |
2 | nfimd.2 | . 2 ⊢ (φ → Ⅎxχ) | |
3 | nfnf1 1790 | . . . 4 ⊢ ℲxℲxψ | |
4 | nfnf1 1790 | . . . 4 ⊢ ℲxℲxχ | |
5 | nfr 1761 | . . . . . 6 ⊢ (Ⅎxχ → (χ → ∀xχ)) | |
6 | 5 | imim2d 48 | . . . . 5 ⊢ (Ⅎxχ → ((ψ → χ) → (ψ → ∀xχ))) |
7 | 19.21t 1795 | . . . . . 6 ⊢ (Ⅎxψ → (∀x(ψ → χ) ↔ (ψ → ∀xχ))) | |
8 | 7 | biimprd 214 | . . . . 5 ⊢ (Ⅎxψ → ((ψ → ∀xχ) → ∀x(ψ → χ))) |
9 | 6, 8 | syl9r 67 | . . . 4 ⊢ (Ⅎxψ → (Ⅎxχ → ((ψ → χ) → ∀x(ψ → χ)))) |
10 | 3, 4, 9 | alrimd 1769 | . . 3 ⊢ (Ⅎxψ → (Ⅎxχ → ∀x((ψ → χ) → ∀x(ψ → χ)))) |
11 | df-nf 1545 | . . 3 ⊢ (Ⅎx(ψ → χ) ↔ ∀x((ψ → χ) → ∀x(ψ → χ))) | |
12 | 10, 11 | syl6ibr 218 | . 2 ⊢ (Ⅎxψ → (Ⅎxχ → Ⅎx(ψ → χ))) |
13 | 1, 2, 12 | sylc 56 | 1 ⊢ (φ → Ⅎx(ψ → χ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 Ⅎwnf 1544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-11 1746 |
This theorem depends on definitions: df-bi 177 df-ex 1542 df-nf 1545 |
This theorem is referenced by: nfimOLD 1814 hbimd 1815 19.23tOLD 1819 nfand 1822 nfbid 1832 nfbidOLD 1833 nfnfOLD 1846 19.21tOLD 1863 dvelimf 1997 nfsb4t 2080 nfmod2 2217 nfrald 2665 nfifd 3685 |
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