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Mirrors > Home > ILE Home > Th. List > r19.2uz | GIF version |
Description: A version of r19.2m 3346 for upper integer quantifiers. (Contributed by Mario Carneiro, 15-Feb-2014.) |
Ref | Expression |
---|---|
rexuz3.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
Ref | Expression |
---|---|
r19.2uz | ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 → ∃𝑘 ∈ 𝑍 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 8745 | . . . . . 6 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ) | |
2 | uzid 8750 | . . . . . 6 ⊢ (𝑗 ∈ ℤ → 𝑗 ∈ (ℤ≥‘𝑗)) | |
3 | elex2 2624 | . . . . . 6 ⊢ (𝑗 ∈ (ℤ≥‘𝑗) → ∃𝑘 𝑘 ∈ (ℤ≥‘𝑗)) | |
4 | 1, 2, 3 | 3syl 17 | . . . . 5 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → ∃𝑘 𝑘 ∈ (ℤ≥‘𝑗)) |
5 | rexuz3.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | 4, 5 | eleq2s 2177 | . . . 4 ⊢ (𝑗 ∈ 𝑍 → ∃𝑘 𝑘 ∈ (ℤ≥‘𝑗)) |
7 | r19.2m 3346 | . . . 4 ⊢ ((∃𝑘 𝑘 ∈ (ℤ≥‘𝑗) ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → ∃𝑘 ∈ (ℤ≥‘𝑗)𝜑) | |
8 | 6, 7 | sylan 277 | . . 3 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → ∃𝑘 ∈ (ℤ≥‘𝑗)𝜑) |
9 | 5 | uztrn2 8753 | . . . . . . 7 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍) |
10 | 9 | ex 113 | . . . . . 6 ⊢ (𝑗 ∈ 𝑍 → (𝑘 ∈ (ℤ≥‘𝑗) → 𝑘 ∈ 𝑍)) |
11 | 10 | anim1d 329 | . . . . 5 ⊢ (𝑗 ∈ 𝑍 → ((𝑘 ∈ (ℤ≥‘𝑗) ∧ 𝜑) → (𝑘 ∈ 𝑍 ∧ 𝜑))) |
12 | 11 | reximdv2 2465 | . . . 4 ⊢ (𝑗 ∈ 𝑍 → (∃𝑘 ∈ (ℤ≥‘𝑗)𝜑 → ∃𝑘 ∈ 𝑍 𝜑)) |
13 | 12 | imp 122 | . . 3 ⊢ ((𝑗 ∈ 𝑍 ∧ ∃𝑘 ∈ (ℤ≥‘𝑗)𝜑) → ∃𝑘 ∈ 𝑍 𝜑) |
14 | 8, 13 | syldan 276 | . 2 ⊢ ((𝑗 ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → ∃𝑘 ∈ 𝑍 𝜑) |
15 | 14 | rexlimiva 2477 | 1 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 → ∃𝑘 ∈ 𝑍 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∃wex 1422 ∈ wcel 1434 ∀wral 2353 ∃wrex 2354 ‘cfv 4953 ℤcz 8468 ℤ≥cuz 8736 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-cnex 7165 ax-resscn 7166 ax-pre-ltirr 7186 ax-pre-ltwlin 7187 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2612 df-sbc 2826 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-br 3807 df-opab 3861 df-mpt 3862 df-id 4077 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-iota 4918 df-fun 4955 df-fn 4956 df-f 4957 df-fv 4961 df-ov 5567 df-pnf 7253 df-mnf 7254 df-xr 7255 df-ltxr 7256 df-le 7257 df-neg 7385 df-z 8469 df-uz 8737 |
This theorem is referenced by: recvguniq 10066 climge0 10348 |
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