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Theorem nfsbd 1950
Description: Deduction version of nfsb 1919. (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 1499 . 2 (𝜑 → ∀𝑥𝜑)
3 nfsbd.2 . . 3 (𝜑 → Ⅎ𝑧𝜓)
43alimi 1431 . 2 (∀𝑥𝜑 → ∀𝑥𝑧𝜓)
5 nfsbt 1949 . 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 1329  wnf 1436  [wsb 1735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  nfeud  2015  nfabd  2300  nfraldya  2469  nfrexdya  2470  cbvrald  13054
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