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Theorem hbsb4 1936
Description: A variable not free remains so after substitution with a distinct variable. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
Hypothesis
Ref Expression
hbsb4.1 (𝜑 → ∀𝑧𝜑)
Assertion
Ref Expression
hbsb4 (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))

Proof of Theorem hbsb4
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 hbsb4.1 . . 3 (𝜑 → ∀𝑧𝜑)
21hbsb 1871 . 2 ([𝑤 / 𝑥]𝜑 → ∀𝑧[𝑤 / 𝑥]𝜑)
3 sbequ 1768 . 2 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
42, 3dvelimALT 1934 1 (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1287  [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-in2 580  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:  hbsb4t  1937  dvelimf  1939
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