Theorem equsalvw

Description: Version of equsalv with a disjoint variable condition, and of equsal with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsexvw . (Contributed by BJ, 31-May-2019)

	Ref	Expression
Hypothesis	equsalvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	equsalvw	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		equsalvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2	1	pm5.74i	$$⊢\left(x=y\to \phi \right)\leftrightarrow \left(x=y\to \psi \right)$$
3	2	albii	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \forall x\left(x=y\to \psi \right)$$
4		equsv	$$⊢\forall x\left(x=y\to \psi \right)\leftrightarrow \psi$$
5	3 4	bitri	$$⊢\forall x\left(x=y\to \phi \right)\leftrightarrow \psi$$