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Metamath Proof Explorer

Theorem reuimrmo

Description: Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo . (Contributed by Alexander van der Vekens, 25-Jun-2017)

		Ref	Expression
	Assertion	reuimrmo	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∃! 𝑥 ∈ 𝐴 𝜓 → ∃* 𝑥 ∈ 𝐴 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		reurmo	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜓 → ∃* 𝑥 ∈ 𝐴 𝜓 )
2		rmoim	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∃* 𝑥 ∈ 𝐴 𝜓 → ∃* 𝑥 ∈ 𝐴 𝜑 ) )
3	1 2	syl5	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∃! 𝑥 ∈ 𝐴 𝜓 → ∃* 𝑥 ∈ 𝐴 𝜑 ) )