![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfreud | Structured version Visualization version GIF version |
Description: Deduction version of nfreu 3315. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 8-Oct-2016.) |
Ref | Expression |
---|---|
nfreud.1 | ⊢ Ⅎ𝑦𝜑 |
nfreud.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfreud.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfreud | ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-reu 3095 | . 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
2 | nfreud.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvf 2958 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
4 | 3 | adantl 474 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦) |
5 | nfreud.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | 5 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴) |
7 | 4, 6 | nfeld 2941 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴) |
8 | nfreud.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
9 | 8 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
10 | 7, 9 | nfand 1860 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
11 | 2, 10 | nfeud2 2610 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
12 | 1, 11 | nfxfrd 1816 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∀wal 1505 Ⅎwnf 1746 ∈ wcel 2050 ∃!weu 2583 Ⅎwnfc 2916 ∃!wreu 3090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-mo 2547 df-eu 2584 df-cleq 2771 df-clel 2846 df-nfc 2918 df-reu 3095 |
This theorem is referenced by: nfreu 3315 |
Copyright terms: Public domain | W3C validator |