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Mirrors > Home > MPE Home > Th. List > nfeud2 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2390. Check out nfeudw 2677 for a version that replaces the distinctor with a disjoint variable condition, not requiring ax-13 2390. (Contributed by Mario Carneiro, 14-Nov-2016.) (Proof shortened by Wolf Lammen, 4-Oct-2018.) (Proof shortened by BJ, 14-Oct-2022.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfeud2.1 | ⊢ Ⅎ𝑦𝜑 |
nfeud2.2 | ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfeud2 | ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 2654 | . 2 ⊢ (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓)) | |
2 | nfeud2.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfeud2.2 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) | |
4 | 2, 3 | nfexd2 2468 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
5 | 2, 3 | nfmod2 2642 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
6 | 4, 5 | nfand 1898 | . 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 ∧ ∃*𝑦𝜓)) |
7 | 1, 6 | nfxfrd 1854 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∀wal 1535 ∃wex 1780 Ⅎwnf 1784 ∃*wmo 2620 ∃!weu 2653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-mo 2622 df-eu 2654 |
This theorem is referenced by: nfeud 2678 nfreud 3372 |
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