Theorem equsexvw

Description: Version of equsexv with a disjoint variable condition, and of equsex with two disjoint variable conditions, which requires fewer axioms. See also the dual form equsalvw . (Contributed by BJ, 31-May-2019) (Proof shortened by Wolf Lammen, 23-Oct-2023)

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

Proof

Step	Hyp	Ref	Expression
1		equsalvw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2		alinexa	$$⊢\forall x\left(x=y\to \neg \phi \right)\leftrightarrow \neg \exists x\left(x=y\wedge \phi \right)$$
3	1	notbid	$$⊢x=y\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
4	3	equsalvw	$$⊢\forall x\left(x=y\to \neg \phi \right)\leftrightarrow \neg \psi$$
5	2 4	bitr3i	$$⊢\neg \exists x\left(x=y\wedge \phi \right)\leftrightarrow \neg \psi$$
6	5	con4bii	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$