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Theorem nfsbd 1988
Description: Deduction version of nfsb 1957. (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 1529 . 2 (𝜑 → ∀𝑥𝜑)
3 nfsbd.2 . . 3 (𝜑 → Ⅎ𝑧𝜓)
43alimi 1465 . 2 (∀𝑥𝜑 → ∀𝑥𝑧𝜓)
5 nfsbt 1987 . 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 1361  wnf 1470  [wsb 1772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773
This theorem is referenced by:  nfeud  2053  nfabd  2351  nfraldya  2524  nfrexdya  2525  cbvrald  14923
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