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Theorem nfsb4or 1942
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 1865 . 2 𝑧[𝑤 / 𝑥]𝜑
3 sbequ 1763 . 2 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
42, 3dvelimor 1937 1 (∀𝑧 𝑧 = 𝑦 ∨ Ⅎ𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wo 662  wal 1283  wnf 1390  [wsb 1687
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by: (None)
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