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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 ∈ 𝐴 → ( 𝜑 → 𝜓 ) )
	Assertion	rmoimia	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 → ∃* 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		rmoimia.1	⊢ ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) )
2		rmoim	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∃* 𝑥 ∈ 𝐴 𝜓 → ∃* 𝑥 ∈ 𝐴 𝜑 ) )
3	2 1	mprg	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 → ∃* 𝑥 ∈ 𝐴 𝜑 )