Theorem bj-hbsb2av

Description: Version of hbsb2a with a disjoint variable condition, which does not require ax-13 . (Contributed by BJ, 11-Sep-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	bj-hbsb2av	⊢ ( [ 𝑦 / 𝑥 ] ∀ 𝑦 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		sb4av	⊢ ( [ 𝑦 / 𝑥 ] ∀ 𝑦 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
2		sb6	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
3	2	biimpri	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → [ 𝑦 / 𝑥 ] 𝜑 )
4	3	axc4i	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
5	1 4	syl	⊢ ( [ 𝑦 / 𝑥 ] ∀ 𝑦 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )