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Mirrors > Home > MPE Home > Th. List > nfreu | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for restricted unique existence. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfreuw 3303 when possible. (Contributed by NM, 30-Oct-2010.) (Revised by Mario Carneiro, 8-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfreu.1 | ⊢ Ⅎ𝑥𝐴 |
nfreu.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfreu | ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1807 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfreu.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfreu.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
6 | 1, 3, 5 | nfreud 3300 | . 2 ⊢ (⊤ → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑) |
7 | 6 | mptru 1546 | 1 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1540 Ⅎwnf 1786 Ⅎwnfc 2887 ∃!wreu 3066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-13 2372 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1542 df-ex 1783 df-nf 1787 df-mo 2540 df-eu 2569 df-cleq 2730 df-clel 2816 df-nfc 2889 df-reu 3071 |
This theorem is referenced by: (None) |
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