Theorem equsexh

Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See equsexhv for a version with a disjoint variable condition which does not require ax-13 . (Contributed by NM, 5-Aug-1993) (New usage is discouraged.)

	Ref	Expression
Hypotheses	equsalh.1	$$⊢\psi \to \forall x\psi$$
	equsalh.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	equsexh	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$

Proof

Step	Hyp	Ref	Expression
1		equsalh.1	$$⊢\psi \to \forall x\psi$$
2		equsalh.2	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3	1	nf5i	$$⊢Ⅎx\psi$$
4	3 2	equsex	$$⊢\exists x\left(x=y\wedge \phi \right)\leftrightarrow \psi$$