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	$$⊢\underset{\_}{Ⅎ}yA$$
	sbcalfi.2	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \psi$$
Assertion	sbcalfi	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\forall y\phi \leftrightarrow \forall y\psi$$

Proof

Step	Hyp	Ref	Expression
1		sbcalfi.1	$$⊢\underset{\_}{Ⅎ}yA$$
2		sbcalfi.2	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \psi$$
3	1	sbcalf	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\forall y\phi \leftrightarrow \forall y\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi$$
4	2	albii	$$⊢\forall y\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\phi \leftrightarrow \forall y\psi$$
5	3 4	bitri	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\forall y\phi \leftrightarrow \forall y\psi$$