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Metamath Proof Explorer

Theorem exlimi

Description: Inference associated with 19.23 . See exlimiv for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 10-Jan-1993) (Revised by Mario Carneiro, 24-Sep-2016)

	Ref	Expression
Hypotheses	exlimi.1	⊢ Ⅎ 𝑥 𝜓
	exlimi.2	⊢ ( 𝜑 → 𝜓 )
Assertion	exlimi	⊢ ( ∃ 𝑥 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		exlimi.1	⊢ Ⅎ 𝑥 𝜓
2		exlimi.2	⊢ ( 𝜑 → 𝜓 )
3	1	19.23	⊢ ( ∀ 𝑥 ( 𝜑 → 𝜓 ) ↔ ( ∃ 𝑥 𝜑 → 𝜓 ) )
4	3 2	mpgbi	⊢ ( ∃ 𝑥 𝜑 → 𝜓 )