Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > rexuz2 | GIF version |
Description: Restricted existential quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.) |
Ref | Expression |
---|---|
rexuz2 | ⊢ (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2 8479 | . . . . . 6 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) | |
2 | df-3an 887 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) ↔ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑀 ≤ 𝑛)) | |
3 | 1, 2 | bitri 173 | . . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) ↔ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑀 ≤ 𝑛)) |
4 | 3 | anbi1i 431 | . . . 4 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝜑) ↔ (((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑀 ≤ 𝑛) ∧ 𝜑)) |
5 | anass 381 | . . . . 5 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑀 ≤ 𝑛) ∧ 𝜑) ↔ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (𝑀 ≤ 𝑛 ∧ 𝜑))) | |
6 | anass 381 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (𝑀 ≤ 𝑛 ∧ 𝜑)) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ (𝑀 ≤ 𝑛 ∧ 𝜑)))) | |
7 | an12 495 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ (𝑀 ≤ 𝑛 ∧ 𝜑))) ↔ (𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ (𝑀 ≤ 𝑛 ∧ 𝜑)))) | |
8 | 6, 7 | bitri 173 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (𝑀 ≤ 𝑛 ∧ 𝜑)) ↔ (𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ (𝑀 ≤ 𝑛 ∧ 𝜑)))) |
9 | 5, 8 | bitri 173 | . . . 4 ⊢ ((((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑀 ≤ 𝑛) ∧ 𝜑) ↔ (𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ (𝑀 ≤ 𝑛 ∧ 𝜑)))) |
10 | 4, 9 | bitri 173 | . . 3 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) ∧ 𝜑) ↔ (𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ (𝑀 ≤ 𝑛 ∧ 𝜑)))) |
11 | 10 | rexbii2 2335 | . 2 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∃𝑛 ∈ ℤ (𝑀 ∈ ℤ ∧ (𝑀 ≤ 𝑛 ∧ 𝜑))) |
12 | r19.42v 2467 | . 2 ⊢ (∃𝑛 ∈ ℤ (𝑀 ∈ ℤ ∧ (𝑀 ≤ 𝑛 ∧ 𝜑)) ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑))) | |
13 | 11, 12 | bitri 173 | 1 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ ∧ ∃𝑛 ∈ ℤ (𝑀 ≤ 𝑛 ∧ 𝜑))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∧ w3a 885 ∈ wcel 1393 ∃wrex 2307 class class class wbr 3764 ‘cfv 4902 ≤ cle 7061 ℤcz 8245 ℤ≥cuz 8473 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-cnex 6975 ax-resscn 6976 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-fv 4910 df-ov 5515 df-neg 7185 df-z 8246 df-uz 8474 |
This theorem is referenced by: 2rexuz 8525 |
Copyright terms: Public domain | W3C validator |