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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	$⊢ φ \to (ψ \to χ)$
Assertion	exlimd	$⊢ φ \to (\exists x ψ \to χ)$

Proof

Step	Hyp	Ref	Expression
1		exlimd.1	$⊢ Ⅎ x φ$
2		exlimd.2	$⊢ Ⅎ x χ$
3		exlimd.3	$⊢ φ \to (ψ \to χ)$
4	1 3	eximd	$⊢ φ \to (\exists x ψ \to \exists x χ)$
5	2	19.9	$⊢ \exists x χ \leftrightarrow χ$
6	4 5	imbitrdi	$⊢ φ \to (\exists x ψ \to χ)$