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Metamath Proof Explorer

Theorem spcev

Description: Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993) (Proof shortened by Eric Schmidt, 22-Dec-2006)

	Ref	Expression
Hypotheses	spcv.1	⊢ 𝐴 ∈ V
	spcv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion	spcev	⊢ ( 𝜓 → ∃ 𝑥 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		spcv.1	⊢ 𝐴 ∈ V
2		spcv.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3	2	spcegv	⊢ ( 𝐴 ∈ V → ( 𝜓 → ∃ 𝑥 𝜑 ) )
4	1 3	ax-mp	⊢ ( 𝜓 → ∃ 𝑥 𝜑 )