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Metamath Proof Explorer

Theorem exlimd

Description: Deduction form of Theorem 19.9 of Margaris p. 89. (Contributed by NM, 23-Jan-1993) (Revised by Mario Carneiro, 24-Sep-2016) (Proof shortened by Wolf Lammen, 12-Jan-2018)

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

Proof

Step	Hyp	Ref	Expression
1		exlimd.1	⊢ Ⅎ 𝑥 𝜑
2		exlimd.2	⊢ Ⅎ 𝑥 𝜒
3		exlimd.3	⊢ ( 𝜑 → ( 𝜓 → 𝜒 ) )
4	1 3	eximd	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 → ∃ 𝑥 𝜒 ) )
5	2	19.9	⊢ ( ∃ 𝑥 𝜒 ↔ 𝜒 )
6	4 5	imbitrdi	⊢ ( 𝜑 → ( ∃ 𝑥 𝜓 → 𝜒 ) )