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Theorem bj-hbsb3v 32396
Description: Version of hbsb3 2368 with a dv condition, which does not require ax-13 2250. (Remark: the unbundled version of nfs1 2369 is given by bj-nfs1v 32394.) (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 1888 . 2 ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
3 bj-hbsb2av 32395 . 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 1478  [wsb 1882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-10 2021  ax-12 2049  ax-13 2250
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1883
This theorem is referenced by: (None)
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