Theorem sbbibvv

Description: Reversal of substitution. (Contributed by AV, 6-Aug-2023)

	Ref	Expression
Assertion	sbbibvv	⊢ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		nfv	⊢ Ⅎ 𝑦 𝜑
2		nfv	⊢ Ⅎ 𝑥 𝜓
3	1 2	sbbib	⊢ ( ∀ 𝑦 ( [ 𝑦 / 𝑥 ] 𝜑 ↔ 𝜓 ) ↔ ∀ 𝑥 ( 𝜑 ↔ [ 𝑥 / 𝑦 ] 𝜓 ) )