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Mirrors > Home > ILE Home > Th. List > fz1sbc | GIF version |
Description: Quantification over a one-member finite set of sequential integers in terms of substitution. (Contributed by NM, 28-Nov-2005.) |
Ref | Expression |
---|---|
fz1sbc | ⊢ (𝑁 ∈ ℤ → (∀𝑘 ∈ (𝑁...𝑁)φ ↔ [𝑁 / 𝑘]φ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbc6g 2782 | . 2 ⊢ (𝑁 ∈ ℤ → ([𝑁 / 𝑘]φ ↔ ∀𝑘(𝑘 = 𝑁 → φ))) | |
2 | df-ral 2305 | . . 3 ⊢ (∀𝑘 ∈ (𝑁...𝑁)φ ↔ ∀𝑘(𝑘 ∈ (𝑁...𝑁) → φ)) | |
3 | elfz1eq 8669 | . . . . . 6 ⊢ (𝑘 ∈ (𝑁...𝑁) → 𝑘 = 𝑁) | |
4 | elfz3 8668 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁)) | |
5 | eleq1 2097 | . . . . . . 7 ⊢ (𝑘 = 𝑁 → (𝑘 ∈ (𝑁...𝑁) ↔ 𝑁 ∈ (𝑁...𝑁))) | |
6 | 4, 5 | syl5ibrcom 146 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑘 = 𝑁 → 𝑘 ∈ (𝑁...𝑁))) |
7 | 3, 6 | impbid2 131 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑘 ∈ (𝑁...𝑁) ↔ 𝑘 = 𝑁)) |
8 | 7 | imbi1d 220 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑘 ∈ (𝑁...𝑁) → φ) ↔ (𝑘 = 𝑁 → φ))) |
9 | 8 | albidv 1702 | . . 3 ⊢ (𝑁 ∈ ℤ → (∀𝑘(𝑘 ∈ (𝑁...𝑁) → φ) ↔ ∀𝑘(𝑘 = 𝑁 → φ))) |
10 | 2, 9 | syl5rbb 182 | . 2 ⊢ (𝑁 ∈ ℤ → (∀𝑘(𝑘 = 𝑁 → φ) ↔ ∀𝑘 ∈ (𝑁...𝑁)φ)) |
11 | 1, 10 | bitr2d 178 | 1 ⊢ (𝑁 ∈ ℤ → (∀𝑘 ∈ (𝑁...𝑁)φ ↔ [𝑁 / 𝑘]φ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1240 = wceq 1242 ∈ wcel 1390 ∀wral 2300 [wsbc 2758 (class class class)co 5455 ℤcz 8021 ...cfz 8644 |
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-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-pre-ltirr 6795 ax-pre-apti 6798 |
This theorem depends on definitions: df-bi 110 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-neg 6982 df-z 8022 df-uz 8250 df-fz 8645 |
This theorem is referenced by: (None) |
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