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Mirrors > Home > MPE Home > Th. List > nfreuw | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for restricted unique existence. Version of nfreu 3375 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 30-Oct-2010.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
nfreuw.1 | ⊢ Ⅎ𝑥𝐴 |
nfreuw.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfreuw | ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 3144 | . . 3 ⊢ (∃!𝑦 ∈ 𝐴 𝜑 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
2 | nftru 1804 | . . . 4 ⊢ Ⅎ𝑦⊤ | |
3 | nfcvd 2977 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝑦) | |
4 | nfreuw.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐴 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑥𝐴) |
6 | 3, 5 | nfeld 2988 | . . . . 5 ⊢ (⊤ → Ⅎ𝑥 𝑦 ∈ 𝐴) |
7 | nfreuw.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (⊤ → Ⅎ𝑥𝜑) |
9 | 6, 8 | nfand 1897 | . . . 4 ⊢ (⊤ → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) |
10 | 2, 9 | nfeudw 2676 | . . 3 ⊢ (⊤ → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) |
11 | 1, 10 | nfxfrd 1853 | . 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑) |
12 | 11 | mptru 1543 | 1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ⊤wtru 1537 Ⅎwnf 1783 ∈ wcel 2113 ∃!weu 2652 Ⅎwnfc 2960 ∃!wreu 3139 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-mo 2621 df-eu 2653 df-cleq 2813 df-clel 2892 df-nfc 2962 df-reu 3144 |
This theorem is referenced by: sbcreu 3853 reuccatpfxs1 14102 2reu7 43385 2reu8 43386 |
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