Theorem sb4e

Description: One direction of a simplified definition of substitution that unlike sb4b does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Feb-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	sb4e	$$⊢\left[y/x\right]\phi \to \forall x\left(x=y\to \exists y\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		sb1	$$⊢\left[y/x\right]\phi \to \exists x\left(x=y\wedge \phi \right)$$
2		equs5e	$$⊢\exists x\left(x=y\wedge \phi \right)\to \forall x\left(x=y\to \exists y\phi \right)$$
3	1 2	syl	$$⊢\left[y/x\right]\phi \to \forall x\left(x=y\to \exists y\phi \right)$$