Theorem bj-hbsb3v 37238

Description: Version of hbsb3 2508 with a disjoint variable condition, which does not require ax-13 2393. (Remark: the unbundled version of nfs1 2509 is given by bj-nfs1v 37236.) (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

Step	Hyp	Ref	Expression
1		bj-hbsb3v.1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
2	1	sbimi 2097	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑)
3		bj-hbsb2av 37237	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
4	2, 3	syl 17	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1548  [wsb 2080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-10 2165  ax-12 2202

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-ex 1790  df-nf 1794  df-sb 2081

This theorem is referenced by: (None)