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Metamath Proof Explorer

Theorem rmoimia

Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017)

		Ref	Expression
	Hypothesis	rmoimia.1	$⊢ x \in A \to (φ \to ψ)$
	Assertion	rmoimia	$⊢ \exists^{} x \in A ψ \to \exists^{} x \in A φ$

Proof

Step	Hyp	Ref	Expression
1		rmoimia.1	$⊢ x \in A \to (φ \to ψ)$
2		rmoim	$⊢ \forall x \in A (φ \to ψ) \to (\exists^{} x \in A ψ \to \exists^{} x \in A φ)$
3	2 1	mprg	$⊢ \exists^{} x \in A ψ \to \exists^{} x \in A φ$