Metamath Proof Explorer

Theorem rmoim

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

		Ref	Expression
	Assertion	rmoim	$⊢ \forall x \in A (φ \to ψ) \to (\exists^{} x \in A ψ \to \exists^{} x \in A φ)$

Proof

Step	Hyp	Ref	Expression
1		df-ral	$⊢ \forall x \in A (φ \to ψ) \leftrightarrow \forall x (x \in A \to (φ \to ψ))$
2		imdistan	$⊢ (x \in A \to (φ \to ψ)) \leftrightarrow ((x \in A \land φ) \to (x \in A \land ψ))$
3	2	albii	$⊢ \forall x (x \in A \to (φ \to ψ)) \leftrightarrow \forall x ((x \in A \land φ) \to (x \in A \land ψ))$
4	1 3	bitri	$⊢ \forall x \in A (φ \to ψ) \leftrightarrow \forall x ((x \in A \land φ) \to (x \in A \land ψ))$
5		moim	$⊢ \forall x ((x \in A \land φ) \to (x \in A \land ψ)) \to (\exists^{} x (x \in A \land ψ) \to \exists^{} x (x \in A \land φ))$
6		df-rmo	$⊢ \exists^{} x \in A ψ \leftrightarrow \exists^{} x (x \in A \land ψ)$
7		df-rmo	$⊢ \exists^{} x \in A φ \leftrightarrow \exists^{} x (x \in A \land φ)$
8	5 6 7	3imtr4g	$⊢ \forall x ((x \in A \land φ) \to (x \in A \land ψ)) \to (\exists^{} x \in A ψ \to \exists^{} x \in A φ)$
9	4 8	sylbi	$⊢ \forall x \in A (φ \to ψ) \to (\exists^{} x \in A ψ \to \exists^{} x \in A φ)$