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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 1467 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | nfor.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfri 1467 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) |
5 | 2, 4 | hbor 1493 | . 2 ⊢ ((𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) |
6 | 5 | nfi 1406 | 1 ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 670 Ⅎwnf 1404 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-gen 1393 ax-4 1455 |
This theorem depends on definitions: df-bi 116 df-nf 1405 |
This theorem is referenced by: nfdc 1605 nfun 3179 nfpr 3520 nfso 4162 nffrec 6223 indpi 7051 nfsum1 10964 nfsum 10965 bj-findis 12762 isomninnlem 12809 |
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