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| Description: Restricted "at most one" in term of uniqueness. (Contributed by NM, 16-Jun-2017.) | 
| Ref | Expression | 
|---|---|
| rmo5 | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | moeu 2582 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | |
| 2 | df-rmo 3379 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 3 | df-rex 3070 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 4 | df-reu 3380 | . . 3 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 5 | 3, 4 | imbi12i 350 | . 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | 
| 6 | 1, 2, 5 | 3bitr4i 303 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 → ∃!𝑥 ∈ 𝐴 𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1778 ∈ wcel 2107 ∃*wmo 2537 ∃!weu 2567 ∃wrex 3069 ∃!wreu 3377 ∃*wrmo 3378 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-mo 2539 df-eu 2568 df-rex 3070 df-rmo 3379 df-reu 3380 | 
| This theorem is referenced by: nrexrmo 3400 cbvrmo 3428 2reurex 3765 rmo0 4361 rmosn 4718 ddemeas 34238 iccpartdisj 47429 | 
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