ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfsbd GIF version

Theorem nfsbd 1977
Description: Deduction version of nfsb 1946. (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 1519 . 2 (𝜑 → ∀𝑥𝜑)
3 nfsbd.2 . . 3 (𝜑 → Ⅎ𝑧𝜓)
43alimi 1455 . 2 (∀𝑥𝜑 → ∀𝑥𝑧𝜓)
5 nfsbt 1976 . 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 1351  wnf 1460  [wsb 1762
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763
This theorem is referenced by:  nfeud  2042  nfabd  2339  nfraldya  2512  nfrexdya  2513  cbvrald  14728
  Copyright terms: Public domain W3C validator