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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 → ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		dfsb	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
2		alequexv	⊢ ( ∀ 𝑦 ( 𝑦 = 𝑡 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) → ∃ 𝑦 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
3	1 2	sylbi	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 → ∃ 𝑦 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
4		exsbim	⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 𝜑 )
5	3 4	syl	⊢ ( [ 𝑡 / 𝑥 ] 𝜑 → ∃ 𝑥 𝜑 )