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Theorem nfsbd 1851
 Description: Deduction version of nfsb 1822. (Contributed by NM, 15-Feb-2013.)
Hypotheses
Ref Expression
nfsbd.1 𝑥𝜑
nfsbd.2 (𝜑 → Ⅎ𝑧𝜓)
Assertion
Ref Expression
nfsbd (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
Distinct variable group:   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem nfsbd
StepHypRef Expression
1 nfsbd.1 . . 3 𝑥𝜑
21nfri 1412 . 2 (𝜑 → ∀𝑥𝜑)
3 nfsbd.2 . . 3 (𝜑 → Ⅎ𝑧𝜓)
43alimi 1344 . 2 (∀𝑥𝜑 → ∀𝑥𝑧𝜓)
5 nfsbt 1850 . 2 (∀𝑥𝑧𝜓 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
62, 4, 53syl 17 1 (𝜑 → Ⅎ𝑧[𝑦 / 𝑥]𝜓)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  Ⅎwnf 1349  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  nfeud  1916  nfabd  2196  nfraldya  2358  nfrexdya  2359  cbvrald  9927
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