Theorem bj-hbsb3t

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

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

Proof

Step	Hyp	Ref	Expression
1		spsbim	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] ∀ 𝑦 𝜑 ) )
2		hbsb2a	⊢ ( [ 𝑦 / 𝑥 ] ∀ 𝑦 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
3	1 2	syl6	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑦 𝜑 ) → ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 ) )