Theorem equs4

Description: Lemma used in proofs of implicit substitution properties. The converse requires either a disjoint variable condition ( sbalex ) or a nonfreeness hypothesis ( equs45f ). Usage of this theorem is discouraged because it depends on ax-13 . See equs4v for a weaker version requiring fewer axioms. (Contributed by NM, 10-May-1993) (Proof shortened by Mario Carneiro, 20-May-2014) (Proof shortened by Wolf Lammen, 5-Feb-2018) (New usage is discouraged.)

	Ref	Expression
Assertion	equs4	$$⊢\forall x\left(x=y\to \phi \right)\to \exists x\left(x=y\wedge \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		ax6e	$$⊢\exists xx=y$$
2		exintr	$$⊢\forall x\left(x=y\to \phi \right)\to \left(\exists xx=y\to \exists x\left(x=y\wedge \phi \right)\right)$$
3	1 2	mpi	$$⊢\forall x\left(x=y\to \phi \right)\to \exists x\left(x=y\wedge \phi \right)$$