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Mirrors > Home > ILE Home > Th. List > nfs1 | GIF version |
Description: If 𝑦 is not free in 𝜑, 𝑥 is not free in [𝑦 / 𝑥]𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) |
Ref | Expression |
---|---|
nfs1.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
nfs1 | ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfs1.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | nfri 1452 | . . 3 ⊢ (𝜑 → ∀𝑦𝜑) |
3 | 2 | hbsb3 1729 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑) |
4 | 3 | nfi 1391 | 1 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 |
Colors of variables: wff set class |
Syntax hints: Ⅎwnf 1389 [wsb 1685 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-11 1437 ax-4 1440 ax-i9 1463 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 |
This theorem is referenced by: sb8 1777 sb8e 1778 |
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