Theorem equs5a

Description: A property related to substitution that unlike equs5 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 . This proof uses ax12 , see equs5aALT for an alternative one using ax-12 but not ax13 . Usage of the weaker equs5av is preferred, which uses ax12v2 , but not ax-13 . (Contributed by NM, 2-Feb-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	equs5a	$$⊢\exists x\left(x=y\wedge \forall y\phi \right)\to \forall x\left(x=y\to \phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfa1	$$⊢Ⅎx\forall x\left(x=y\to \phi \right)$$
2		ax12	$$⊢x=y\to \left(\forall y\phi \to \forall x\left(x=y\to \phi \right)\right)$$
3	2	imp	$$⊢\left(x=y\wedge \forall y\phi \right)\to \forall x\left(x=y\to \phi \right)$$
4	1 3	exlimi	$$⊢\exists x\left(x=y\wedge \forall y\phi \right)\to \forall x\left(x=y\to \phi \right)$$