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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 1506 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | nfor.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
4 | 3 | nfri 1506 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) |
5 | 2, 4 | hbor 1533 | . 2 ⊢ ((𝜑 ∨ 𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) |
6 | 5 | nfi 1449 | 1 ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓) |
Colors of variables: wff set class |
Syntax hints: ∨ wo 698 Ⅎwnf 1447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-gen 1436 ax-4 1497 |
This theorem depends on definitions: df-bi 116 df-nf 1448 |
This theorem is referenced by: nfdc 1646 nfun 3273 nfpr 3620 nfso 4274 nffrec 6355 indpi 7274 nfsum1 11283 nfsum 11284 nfcprod1 11481 nfcprod 11482 bj-findis 13696 isomninnlem 13743 trirec0 13757 |
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