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Mirrors > Home > NFE Home > Th. List > nfrmo | GIF version |
Description: Bound-variable hypothesis builder for restricted uniqueness. (Contributed by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
nfreu.1 | ⊢ ℲxA |
nfreu.2 | ⊢ Ⅎxφ |
Ref | Expression |
---|---|
nfrmo | ⊢ Ⅎx∃*y ∈ A φ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rmo 2622 | . 2 ⊢ (∃*y ∈ A φ ↔ ∃*y(y ∈ A ∧ φ)) | |
2 | nftru 1554 | . . . 4 ⊢ Ⅎy ⊤ | |
3 | nfcvf 2511 | . . . . . . 7 ⊢ (¬ ∀x x = y → Ⅎxy) | |
4 | nfreu.1 | . . . . . . . 8 ⊢ ℲxA | |
5 | 4 | a1i 10 | . . . . . . 7 ⊢ (¬ ∀x x = y → ℲxA) |
6 | 3, 5 | nfeld 2504 | . . . . . 6 ⊢ (¬ ∀x x = y → Ⅎx y ∈ A) |
7 | nfreu.2 | . . . . . . 7 ⊢ Ⅎxφ | |
8 | 7 | a1i 10 | . . . . . 6 ⊢ (¬ ∀x x = y → Ⅎxφ) |
9 | 6, 8 | nfand 1822 | . . . . 5 ⊢ (¬ ∀x x = y → Ⅎx(y ∈ A ∧ φ)) |
10 | 9 | adantl 452 | . . . 4 ⊢ (( ⊤ ∧ ¬ ∀x x = y) → Ⅎx(y ∈ A ∧ φ)) |
11 | 2, 10 | nfmod2 2217 | . . 3 ⊢ ( ⊤ → Ⅎx∃*y(y ∈ A ∧ φ)) |
12 | 11 | trud 1323 | . 2 ⊢ Ⅎx∃*y(y ∈ A ∧ φ) |
13 | 1, 12 | nfxfr 1570 | 1 ⊢ Ⅎx∃*y ∈ A φ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 358 ⊤ wtru 1316 ∀wal 1540 Ⅎwnf 1544 ∈ wcel 1710 ∃*wmo 2205 Ⅎwnfc 2476 ∃*wrmo 2617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rmo 2622 |
This theorem is referenced by: 2rmorex 3040 |
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