Theorem bj-hbsb3t 34707

Description: A theorem close to a closed form of hbsb3 2490. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-hbsb3t	⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Proof of Theorem bj-hbsb3t

Step	Hyp	Ref	Expression
1		spsbim 2078	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑))
2		hbsb2a 2487	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3	1, 2	syl6 35	1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4  ∀wal 1541  [wsb 2070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2141  ax-12 2175  ax-13 2371

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ex 1788  df-nf 1792  df-sb 2071

This theorem is referenced by:  bj-hbsb3  34708  bj-nfs1t  34709