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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 ( ∀ 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 ) )