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Metamath Proof Explorer

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	$⊢ [t/ x] φ \to \exists x φ$

Proof

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