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Mirrors > Home > ILE Home > Th. List > hbsb | GIF version |
Description: If 𝑧 is not free in 𝜑, it is not free in [𝑦 / 𝑥]𝜑 when 𝑦 and 𝑧 are distinct. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 22-Mar-2018.) |
Ref | Expression |
---|---|
hbsb.1 | ⊢ (𝜑 → ∀𝑧𝜑) |
Ref | Expression |
---|---|
hbsb | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbsb.1 | . . . 4 ⊢ (𝜑 → ∀𝑧𝜑) | |
2 | 1 | nfi 1391 | . . 3 ⊢ Ⅎ𝑧𝜑 |
3 | 2 | nfsb 1863 | . 2 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 |
4 | 3 | nfri 1452 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 [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-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 |
This theorem is referenced by: sb10f 1912 hbsb4 1929 sb8euh 1964 hbab 2072 hblem 2186 |
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