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Metamath Proof Explorer

Theorem rexlimdv

Description: Inference from Theorem 19.23 of Margaris p. 90 (restricted quantifier version). (Contributed by NM, 14-Nov-2002) (Proof shortened by Eric Schmidt, 22-Dec-2006) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020)

		Ref	Expression
	Hypothesis	rexlimdv.1	$⊢ φ \to (x \in A \to (ψ \to χ))$
	Assertion	rexlimdv	$⊢ φ \to (\exists x \in A ψ \to χ)$

Proof

Step	Hyp	Ref	Expression
1		rexlimdv.1	$⊢ φ \to (x \in A \to (ψ \to χ))$
2	1	com3l	$⊢ x \in A \to (ψ \to (φ \to χ))$
3	2	rexlimiv	$⊢ \exists x \in A ψ \to (φ \to χ)$
4	3	com12	$⊢ φ \to (\exists x \in A ψ \to χ)$