Theorem spsbe

Description: Existential generalization: if a proposition is true for a specific instance, then there exists an instance where it is true. (Contributed by NM, 29-Jun-1993) (Proof shortened by Wolf Lammen, 3-May-2018) Revise df-sb . (Revised by BJ, 22-Dec-2020) (Proof shortened by Steven Nguyen, 11-Jul-2023)

	Ref	Expression
Assertion	spsbe	$$⊢\left[t/x\right]\phi \to \exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		dfsb	$$⊢\left[t/x\right]\phi \leftrightarrow \forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)$$
2		alequexv	$$⊢\forall y\left(y=t\to \forall x\left(x=y\to \phi \right)\right)\to \exists y\forall x\left(x=y\to \phi \right)$$
3	1 2	sylbi	$$⊢\left[t/x\right]\phi \to \exists y\forall x\left(x=y\to \phi \right)$$
4		exsbim	$$⊢\exists y\forall x\left(x=y\to \phi \right)\to \exists x\phi$$
5	3 4	syl	$$⊢\left[t/x\right]\phi \to \exists x\phi$$