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Metamath Proof Explorer

Theorem exlimexi

Description: Inference similar to Theorem 19.23 of Margaris p. 90. (Contributed by Alan Sare, 21-Apr-2013) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		exlimexi.1	⊢ ( 𝜓 → ∀ 𝑥 𝜓 )
2		exlimexi.2	⊢ ( ∃ 𝑥 𝜑 → ( 𝜑 → 𝜓 ) )
3		hbe1	⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 ∃ 𝑥 𝜑 )
4	3 1 2	exlimdh	⊢ ( ∃ 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → 𝜓 ) )
5	4	pm2.43i	⊢ ( ∃ 𝑥 𝜑 → 𝜓 )