ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfsb4or GIF version

Theorem nfsb4or 1947
Description: A variable not free remains so after substitution with a distinct variable. (Contributed by Jim Kingdon, 11-May-2018.)
Hypothesis
Ref Expression
nfsb4or.1 𝑧𝜑
Assertion
Ref Expression
nfsb4or (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑥]𝜑)

Proof of Theorem nfsb4or
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 nfsb4or.1 . . 3 𝑧𝜑
21nfsb 1870 . 2 𝑧[𝑤 / 𝑥]𝜑
3 sbequ 1768 . 2 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
42, 3dvelimor 1942 1 (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wo 664  wal 1287  wnf 1394  [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator