Theorem sbhb

Description: Two ways of expressing " x is (effectively) not free in ph ". Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 29-May-2009) (New usage is discouraged.)

	Ref	Expression
Assertion	sbhb	⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑦 ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nfv	⊢ Ⅎ 𝑦 𝜑
2	1	sb8	⊢ ( ∀ 𝑥 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 )
3	2	imbi2i	⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ( 𝜑 → ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) )
4		19.21v	⊢ ( ∀ 𝑦 ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) ↔ ( 𝜑 → ∀ 𝑦 [ 𝑦 / 𝑥 ] 𝜑 ) )
5	3 4	bitr4i	⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ∀ 𝑦 ( 𝜑 → [ 𝑦 / 𝑥 ] 𝜑 ) )