Theorem bj-hbsb3t 31899

Description: A theorem close to a closed form of hbsb3 2352. (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 2382	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑))
2		hbsb2a 2349	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3	1, 2	syl6 34	1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4  ∀wal 1473  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868

This theorem is referenced by:  bj-hbsb3  31900  bj-nfs1t  31901