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Mirrors > Home > MPE Home > Th. List > nfeudw | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2679. Version of nfeud 2677 with a disjoint variable condition, which does not require ax-13 2389. (Contributed by NM, 15-Feb-2013.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
nfeudw.1 | ⊢ Ⅎ𝑦𝜑 |
nfeudw.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfeudw | ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 2653 | . 2 ⊢ (∃!𝑦𝜓 ↔ (∃𝑦𝜓 ∧ ∃*𝑦𝜓)) | |
2 | nfeudw.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfeudw.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
4 | 2, 3 | nfexd 2347 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓) |
5 | 2, 3 | nfmodv 2642 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓) |
6 | 4, 5 | nfand 1897 | . 2 ⊢ (𝜑 → Ⅎ𝑥(∃𝑦𝜓 ∧ ∃*𝑦𝜓)) |
7 | 1, 6 | nfxfrd 1853 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∃wex 1779 Ⅎwnf 1783 ∃*wmo 2619 ∃!weu 2652 |
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-10 2144 ax-11 2160 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 df-mo 2621 df-eu 2653 |
This theorem is referenced by: nfeuw 2678 nfreuw 3373 |
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