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Mirrors > Home > MPE Home > Th. List > reuimrmo | Structured version Visualization version GIF version |
Description: Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2604. (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 3371 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜓) | |
2 | rmoim 3728 | . 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 3053 ∃!wreu 3366 ∃*wrmo 3367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-mo 2526 df-eu 2555 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 |
This theorem is referenced by: 2reurmo 3747 |
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