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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 3800 | . 2 ⊢ (𝑁 ∈ ℤ → ([𝑁 / 𝑘]𝜑 ↔ ∀𝑘(𝑘 = 𝑁 → 𝜑))) | |
2 | df-ral 3143 | . . 3 ⊢ (∀𝑘 ∈ (𝑁...𝑁)𝜑 ↔ ∀𝑘(𝑘 ∈ (𝑁...𝑁) → 𝜑)) | |
3 | elfz1eq 12912 | . . . . . 6 ⊢ (𝑘 ∈ (𝑁...𝑁) → 𝑘 = 𝑁) | |
4 | elfz3 12911 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (𝑁...𝑁)) | |
5 | eleq1 2900 | . . . . . . 7 ⊢ (𝑘 = 𝑁 → (𝑘 ∈ (𝑁...𝑁) ↔ 𝑁 ∈ (𝑁...𝑁))) | |
6 | 4, 5 | syl5ibrcom 249 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑘 = 𝑁 → 𝑘 ∈ (𝑁...𝑁))) |
7 | 3, 6 | impbid2 228 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (𝑘 ∈ (𝑁...𝑁) ↔ 𝑘 = 𝑁)) |
8 | 7 | imbi1d 344 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑘 ∈ (𝑁...𝑁) → 𝜑) ↔ (𝑘 = 𝑁 → 𝜑))) |
9 | 8 | albidv 1917 | . . 3 ⊢ (𝑁 ∈ ℤ → (∀𝑘(𝑘 ∈ (𝑁...𝑁) → 𝜑) ↔ ∀𝑘(𝑘 = 𝑁 → 𝜑))) |
10 | 2, 9 | syl5rbb 286 | . 2 ⊢ (𝑁 ∈ ℤ → (∀𝑘(𝑘 = 𝑁 → 𝜑) ↔ ∀𝑘 ∈ (𝑁...𝑁)𝜑)) |
11 | 1, 10 | bitr2d 282 | 1 ⊢ (𝑁 ∈ ℤ → (∀𝑘 ∈ (𝑁...𝑁)𝜑 ↔ [𝑁 / 𝑘]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 = wceq 1533 ∈ wcel 2110 ∀wral 3138 [wsbc 3771 (class class class)co 7150 ℤcz 11975 ...cfz 12886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-neg 10867 df-z 11976 df-uz 12238 df-fz 12887 |
This theorem is referenced by: (None) |
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