Theorem hbsb2e

Description: Special case of a bound-variable hypothesis builder for substitution. 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	hbsb2e	$$⊢\left[y/x\right]\phi \to \forall x\left[y/x\right]\exists y\phi$$

Proof

Step	Hyp	Ref	Expression
1		sb4e	$$⊢\left[y/x\right]\phi \to \forall x\left(x=y\to \exists y\phi \right)$$
2		sb2	$$⊢\forall x\left(x=y\to \exists y\phi \right)\to \left[y/x\right]\exists y\phi$$
3	2	axc4i	$$⊢\forall x\left(x=y\to \exists y\phi \right)\to \forall x\left[y/x\right]\exists y\phi$$
4	1 3	syl	$$⊢\left[y/x\right]\phi \to \forall x\left[y/x\right]\exists y\phi$$