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	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∃* 𝑥 ∈ 𝐴 𝜓 → ∃* 𝑥 ∈ 𝐴 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) )
2		imdistan	⊢ ( ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) )
3	2	albii	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ( 𝜑 → 𝜓 ) ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) )
4	1 3	bitri	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) ↔ ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) )
5		moim	⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) → ( ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) → ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) )
6		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜓 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) )
7		df-rmo	⊢ ( ∃* 𝑥 ∈ 𝐴 𝜑 ↔ ∃* 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
8	5 6 7	3imtr4g	⊢ ( ∀ 𝑥 ( ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( 𝑥 ∈ 𝐴 ∧ 𝜓 ) ) → ( ∃* 𝑥 ∈ 𝐴 𝜓 → ∃* 𝑥 ∈ 𝐴 𝜑 ) )
9	4 8	sylbi	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝜑 → 𝜓 ) → ( ∃* 𝑥 ∈ 𝐴 𝜓 → ∃* 𝑥 ∈ 𝐴 𝜑 ) )