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Metamath Proof Explorer

Theorem spei

Description: Inference from existential specialization, using implicit substitution. Remove a distinct variable constraint. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker speiv if possible. (Contributed by NM, 19-Aug-1993) (Proof shortened by Wolf Lammen, 12-May-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	spei.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
	spei.2	⊢ 𝜓
Assertion	spei	⊢ ∃ 𝑥 𝜑

Proof

Step	Hyp	Ref	Expression
1		spei.1	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2		spei.2	⊢ 𝜓
3		ax6e	⊢ ∃ 𝑥 𝑥 = 𝑦
4	2 1	mpbiri	⊢ ( 𝑥 = 𝑦 → 𝜑 )
5	3 4	eximii	⊢ ∃ 𝑥 𝜑