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Theorem bj-hbsb3v 34555
 Description: Version of hbsb3 2505 with a disjoint variable condition, which does not require ax-13 2379. (Remark: the unbundled version of nfs1 2506 is given by bj-nfs1v 34553.) (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-hbsb3v.1 (𝜑 → ∀𝑦𝜑)
Assertion
Ref Expression
bj-hbsb3v ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-hbsb3v
StepHypRef Expression
1 bj-hbsb3v.1 . . 3 (𝜑 → ∀𝑦𝜑)
21sbimi 2079 . 2 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
3 bj-hbsb2av 34554 . 2 ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
42, 3syl 17 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by: (None)
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