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| Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2593. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker nfeudw 2590 when possible. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nfeud.1 | ⊢ Ⅎ𝑦𝜑 | 
| nfeud.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) | 
| Ref | Expression | 
|---|---|
| nfeud | ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfeud.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
| 2 | nfeud.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
| 3 | 2 | adantr 480 | . 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | 
| 4 | 1, 3 | nfeud2 2589 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1537 Ⅎwnf 1782 ∃!weu 2567 | 
| 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 ax-7 2006 ax-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-nf 1783 df-mo 2539 df-eu 2568 | 
| This theorem is referenced by: nfeu 2593 | 
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