Theorem equsv

Description: If a formula does not contain a variable x , then it is equivalent to the corresponding prototype of substitution with a fresh variable (see sb6 ). (Contributed by BJ, 23-Jul-2023)

	Ref	Expression
Assertion	equsv	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \phi$$

Proof

Step	Hyp	Ref	Expression
1		19.23v	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \left(\exists xx=y\to \phi \right)$$
2		ax6ev	$$⊢\exists xx=y$$
3	2	a1bi	$$⊢\phi \leftrightarrow \left(\exists xx=y\to \phi \right)$$
4	1 3	bitr4i	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \phi$$