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Mirrors > Home > ILE Home > Th. List > nfsbd | GIF version |
Description: Deduction version of nfsb 1877. (Contributed by NM, 15-Feb-2013.) |
Ref | Expression |
---|---|
nfsbd.1 | ⊢ Ⅎ𝑥𝜑 |
nfsbd.2 | ⊢ (𝜑 → Ⅎ𝑧𝜓) |
Ref | Expression |
---|---|
nfsbd | ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfsbd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | nfri 1464 | . 2 ⊢ (𝜑 → ∀𝑥𝜑) |
3 | nfsbd.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑧𝜓) | |
4 | 3 | alimi 1396 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥Ⅎ𝑧𝜓) |
5 | nfsbt 1905 | . 2 ⊢ (∀𝑥Ⅎ𝑧𝜓 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) | |
6 | 2, 4, 5 | 3syl 17 | 1 ⊢ (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1294 Ⅎwnf 1401 [wsb 1699 |
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 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 |
This theorem depends on definitions: df-bi 116 df-nf 1402 df-sb 1700 |
This theorem is referenced by: nfeud 1971 nfabd 2254 nfraldya 2423 nfrexdya 2424 cbvrald 12396 |
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