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Metamath Proof Explorer

Theorem rexlimd

Description: Deduction form of rexlimd . For a version based on fewer axioms see rexlimdv . (Contributed by NM, 27-May-1998) (Proof shortened by Andrew Salmon, 30-May-2011) (Proof shortened by Wolf Lammen, 14-Jan-2020)

	Ref	Expression
Hypotheses	rexlimd.1	$⊢ Ⅎ x φ$
	rexlimd.2	$⊢ Ⅎ x χ$
	rexlimd.3	$⊢ φ \to (x \in A \to (ψ \to χ))$
Assertion	rexlimd	$⊢ φ \to (\exists x \in A ψ \to χ)$

Proof

Step	Hyp	Ref	Expression
1		rexlimd.1	$⊢ Ⅎ x φ$
2		rexlimd.2	$⊢ Ⅎ x χ$
3		rexlimd.3	$⊢ φ \to (x \in A \to (ψ \to χ))$
4	2	a1i	$⊢ φ \to Ⅎ x χ$
5	1 4 3	rexlimd2	$⊢ φ \to (\exists x \in A ψ \to χ)$