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Mirrors > Home > MPE Home > Th. List > nfreud | Structured version Visualization version GIF version |
Description: Deduction version of nfreu 3093. (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 2903 | . 2 ⊢ (∃!𝑦 ∈ 𝐴 𝜓 ↔ ∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
2 | nfreud.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvf 2774 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
4 | 3 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦) |
5 | nfreud.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴) |
7 | 4, 6 | nfeld 2759 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴) |
8 | nfreud.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
10 | 7, 9 | nfand 1814 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
11 | 2, 10 | nfeud2 2470 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
12 | 1, 11 | nfxfrd 1772 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∀wal 1473 Ⅎwnf 1699 ∈ wcel 1977 ∃!weu 2458 Ⅎwnfc 2738 ∃!wreu 2898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-eu 2462 df-cleq 2603 df-clel 2606 df-nfc 2740 df-reu 2903 |
This theorem is referenced by: nfreu 3093 |
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