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Mirrors > Home > MPE Home > Th. List > fz1sbc | Structured version Visualization version 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 3806 | . 2 ⊢ (𝑁 ∈ ℤ → ([𝑁 / 𝑘]𝜑 ↔ ∀𝑘(𝑘 = 𝑁 → 𝜑))) | |
2 | df-ral 3059 | . . 3 ⊢ (∀𝑘 ∈ (𝑁...𝑁)𝜑 ↔ ∀𝑘(𝑘 ∈ (𝑁...𝑁) → 𝜑)) | |
3 | elfz1eq 13545 | . . . . . 6 ⊢ (𝑘 ∈ (𝑁...𝑁) → 𝑘 = 𝑁) | |
4 | elfz3 13544 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁)) | |
5 | eleq1 2817 | . . . . . . 7 ⊢ (𝑘 = 𝑁 → (𝑘 ∈ (𝑁...𝑁) ↔ 𝑁 ∈ (𝑁...𝑁))) | |
6 | 4, 5 | syl5ibrcom 246 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑘 = 𝑁 → 𝑘 ∈ (𝑁...𝑁))) |
7 | 3, 6 | impbid2 225 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑘 ∈ (𝑁...𝑁) ↔ 𝑘 = 𝑁)) |
8 | 7 | imbi1d 341 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑘 ∈ (𝑁...𝑁) → 𝜑) ↔ (𝑘 = 𝑁 → 𝜑))) |
9 | 8 | albidv 1916 | . . 3 ⊢ (𝑁 ∈ ℤ → (∀𝑘(𝑘 ∈ (𝑁...𝑁) → 𝜑) ↔ ∀𝑘(𝑘 = 𝑁 → 𝜑))) |
10 | 2, 9 | bitr2id 284 | . 2 ⊢ (𝑁 ∈ ℤ → (∀𝑘(𝑘 = 𝑁 → 𝜑) ↔ ∀𝑘 ∈ (𝑁...𝑁)𝜑)) |
11 | 1, 10 | bitr2d 280 | 1 ⊢ (𝑁 ∈ ℤ → (∀𝑘 ∈ (𝑁...𝑁)𝜑 ↔ [𝑁 / 𝑘]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1532 = wceq 1534 ∈ wcel 2099 ∀wral 3058 [wsbc 3776 (class class class)co 7420 ℤcz 12589 ...cfz 13517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-pre-lttri 11213 ax-pre-lttrn 11214 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-neg 11478 df-z 12590 df-uz 12854 df-fz 13518 |
This theorem is referenced by: (None) |
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