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Theorem hbsb4 1888
 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 1823 . 2 ([𝑤 / 𝑥]𝜑 → ∀𝑧[𝑤 / 𝑥]𝜑)
3 sbequ 1721 . 2 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
42, 3dvelimALT 1886 1 (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1241  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  hbsb4t  1889  dvelimf  1891
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