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Mirrors > Home > ILE Home > Th. List > nfor | GIF version |
Description: If x is not free in φ and ψ, it is not free in (φ ∨ ψ). (Contributed by Jim Kingdon, 11-Mar-2018.) |
Ref | Expression |
---|---|
nfor.1 | ⊢ Ⅎxφ |
nfor.2 | ⊢ Ⅎxψ |
Ref | Expression |
---|---|
nfor | ⊢ Ⅎx(φ ∨ ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfor.1 | . . . 4 ⊢ Ⅎxφ | |
2 | 1 | nfri 1409 | . . 3 ⊢ (φ → ∀xφ) |
3 | nfor.2 | . . . 4 ⊢ Ⅎxψ | |
4 | 3 | nfri 1409 | . . 3 ⊢ (ψ → ∀xψ) |
5 | 2, 4 | hbor 1435 | . 2 ⊢ ((φ ∨ ψ) → ∀x(φ ∨ ψ)) |
6 | 5 | nfi 1348 | 1 ⊢ Ⅎx(φ ∨ ψ) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 628 Ⅎwnf 1346 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-gen 1335 ax-4 1397 |
This theorem depends on definitions: df-bi 110 df-nf 1347 |
This theorem is referenced by: nfdc 1546 nfun 3093 nfpr 3411 nfso 4030 nffrec 5921 indpi 6326 bj-findis 9439 |
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