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| Description: Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker nfreuw 3414 when possible. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nfrmo.1 | ⊢ Ⅎ𝑥𝐴 | 
| nfrmo.2 | ⊢ Ⅎ𝑥𝜑 | 
| Ref | Expression | 
|---|---|
| nfreu | ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nftru 1804 | . . 3 ⊢ Ⅎ𝑦⊤ | |
| 2 | nfrmo.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) | 
| 4 | nfrmo.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) | 
| 6 | 1, 3, 5 | nfreud 3433 | . 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑) | 
| 7 | 6 | mptru 1547 | 1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ⊤wtru 1541 Ⅎwnf 1783 Ⅎwnfc 2890 ∃!wreu 3378 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 df-eu 2569 df-cleq 2729 df-clel 2816 df-nfc 2892 df-reu 3381 | 
| This theorem is referenced by: (None) | 
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