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Metamath Proof Explorer

Theorem exinst

Description: Existential Instantiation. Virtual deduction form of exlimexi . (Contributed by Alan Sare, 21-Apr-2013) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	exinst.1	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
	exinst.2	⊢ ( ∃ 𝑥 𝜑 , 𝜑 ▶ 𝜓 )
Assertion	exinst	⊢ ( ∃ 𝑥 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		exinst.1	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
2		exinst.2	⊢ ( ∃ 𝑥 𝜑 , 𝜑 ▶ 𝜓 )
3	2	dfvd2i	⊢ ( ∃ 𝑥 𝜑 → ( 𝜑 → 𝜓 ) )
4	1 3	exlimexi	⊢ ( ∃ 𝑥 𝜑 → 𝜓 )