![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfrmo | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker nfrmow 3328 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfreu.1 | ⊢ Ⅎ𝑥𝐴 |
nfreu.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfrmo | ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rmo 3114 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
2 | nftru 1806 | . . . 4 ⊢ Ⅎ𝑦⊤ | |
3 | nfcvf 2981 | . . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
4 | nfreu.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐴) |
6 | 3, 5 | nfeld 2966 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
7 | nfreu.2 | . . . . . . 7 ⊢ Ⅎ𝑥𝜑 | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑) |
9 | 6, 8 | nfand 1898 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) |
10 | 9 | adantl 485 | . . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) |
11 | 2, 10 | nfmod2 2617 | . . 3 ⊢ (⊤ → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) |
12 | 11 | mptru 1545 | . 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑) |
13 | 1, 12 | nfxfr 1854 | 1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ∀wal 1536 ⊤wtru 1539 Ⅎwnf 1785 ∈ wcel 2111 ∃*wmo 2596 Ⅎwnfc 2936 ∃*wrmo 3109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-13 2379 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-mo 2598 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rmo 3114 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |