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Theorem nfsbd 1906
Description: Deduction version of nfsb 1877. (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 1464 . 2 (𝜑 → ∀𝑥𝜑)
3 nfsbd.2 . . 3 (𝜑 → Ⅎ𝑧𝜓)
43alimi 1396 . 2 (∀𝑥𝜑 → ∀𝑥𝑧𝜓)
5 nfsbt 1905 . 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 1294  wnf 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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