Theorem sbcalfi

Description: Move universal quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019)

	Ref	Expression
Hypotheses	sbcalfi.1	⊢ Ⅎ 𝑦 𝐴
	sbcalfi.2	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜓 )
Assertion	sbcalfi	⊢ ( [ 𝐴 / 𝑥 ] ∀ 𝑦 𝜑 ↔ ∀ 𝑦 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		sbcalfi.1	⊢ Ⅎ 𝑦 𝐴
2		sbcalfi.2	⊢ ( [ 𝐴 / 𝑥 ] 𝜑 ↔ 𝜓 )
3	1	sbcalf	⊢ ( [ 𝐴 / 𝑥 ] ∀ 𝑦 𝜑 ↔ ∀ 𝑦 [ 𝐴 / 𝑥 ] 𝜑 )
4	2	albii	⊢ ( ∀ 𝑦 [ 𝐴 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 𝜓 )
5	3 4	bitri	⊢ ( [ 𝐴 / 𝑥 ] ∀ 𝑦 𝜑 ↔ ∀ 𝑦 𝜓 )