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Metamath Proof Explorer

Theorem exlimimdd

Description: Existential elimination rule of natural deduction. (Contributed by ML, 17-Jul-2020) Shorten exlimdd . (Revised by Wolf Lammen, 3-Sep-2023)

	Ref	Expression
Hypotheses	exlimdd.1	⊢ Ⅎ 𝑥 𝜑
	exlimdd.2	⊢ Ⅎ 𝑥 𝜒
	exlimdd.3	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
	exlimimdd.4	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
Assertion	exlimimdd	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		exlimdd.1	⊢ Ⅎ 𝑥 𝜑
2		exlimdd.2	⊢ Ⅎ 𝑥 𝜒
3		exlimdd.3	⊢ ( 𝜑 → ∃ 𝑥 𝜓 )
4		exlimimdd.4	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
5	1 2 4	exlimd	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 → 𝜒 ) )
6	3 5	mpd	⊢ ( 𝜑 → 𝜒 )