Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfrmod | Structured version Visualization version GIF version |
Description: Deduction version of nfrmo 3307. Usage of this theorem is discouraged because it depends on ax-13 2372. (Contributed by NM, 17-Jun-2017.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfreud.1 | ⊢ Ⅎ𝑦𝜑 |
nfreud.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfreud.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfrmod | ⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rmo 3072 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
2 | nfreud.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvf 2936 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
4 | 3 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦) |
5 | nfreud.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | 5 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴) |
7 | 4, 6 | nfeld 2918 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴) |
8 | nfreud.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
9 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
10 | 7, 9 | nfand 1900 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
11 | 2, 10 | nfmod2 2558 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
12 | 1, 11 | nfxfrd 1856 | 1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∀wal 1537 Ⅎwnf 1786 ∈ wcel 2106 ∃*wmo 2538 Ⅎwnfc 2887 ∃*wrmo 3067 |
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-cleq 2730 df-clel 2816 df-nfc 2889 df-rmo 3072 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |