Theorem equs5e

Description: A property related to substitution that unlike equs5 does not require a distinctor antecedent. This proof uses ax12 , see equs5eALT for an alternative one using ax-12 but not ax13 . Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Feb-2007) (Proof shortened by Wolf Lammen, 15-Jan-2018) (New usage is discouraged.)

	Ref	Expression
Assertion	equs5e	$$⊢\exists x\left(x=y\wedge \phi \right)\to \forall x\left(x=y\to \exists y\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfa1	$$⊢Ⅎx\forall x\left(x=y\to \exists y\phi \right)$$
2		ax12	$$⊢x=y\to \left(\forall y\exists y\phi \to \forall x\left(x=y\to \exists y\phi \right)\right)$$
3		hbe1	$$⊢\exists y\phi \to \forall y\exists y\phi$$
4	3	19.23bi	$$⊢\phi \to \forall y\exists y\phi$$
5	2 4	impel	$$⊢\left(x=y\wedge \phi \right)\to \forall x\left(x=y\to \exists y\phi \right)$$
6	1 5	exlimi	$$⊢\exists x\left(x=y\wedge \phi \right)\to \forall x\left(x=y\to \exists y\phi \right)$$