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Mirrors > Home > NFE Home > Th. List > hban | 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 Wolf Lammen, 2-Jan-2018.) |
Ref | Expression |
---|---|
hb.1 | ⊢ (φ → ∀xφ) |
hb.2 | ⊢ (ψ → ∀xψ) |
Ref | Expression |
---|---|
hban | ⊢ ((φ ∧ ψ) → ∀x(φ ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hb.1 | . . . 4 ⊢ (φ → ∀xφ) | |
2 | 1 | nfi 1551 | . . 3 ⊢ Ⅎxφ |
3 | hb.2 | . . . 4 ⊢ (ψ → ∀xψ) | |
4 | 3 | nfi 1551 | . . 3 ⊢ Ⅎxψ |
5 | 2, 4 | nfan 1824 | . 2 ⊢ Ⅎx(φ ∧ ψ) |
6 | 5 | nfri 1762 | 1 ⊢ ((φ ∧ ψ) → ∀x(φ ∧ ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∀wal 1540 |
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-an 360 df-tru 1319 df-ex 1542 df-nf 1545 |
This theorem is referenced by: hb3anOLD 1831 dvelimv 1939 dvelimh 1964 dvelimALT 2133 dvelimf-o 2180 ax11indalem 2197 ax11inda2ALT 2198 cleqh 2450 hboprab 5562 |
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