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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > reuimrmo | Structured version Visualization version GIF version |
Description: Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2704. (Contributed by Alexander van der Vekens, 25-Jun-2017.) |
Ref | Expression |
---|---|
reuimrmo | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reurmo 3373 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜓) | |
2 | rmoim 3634 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) | |
3 | 1, 2 | syl5 34 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wral 3117 ∃!wreu 3119 ∃*wrmo 3120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1879 df-mo 2605 df-eu 2640 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 |
This theorem is referenced by: 2reurmo 42001 |
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