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Mirrors > Home > MPE Home > Th. List > raluz2 | Structured version Visualization version GIF version |
Description: Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.) |
Ref | Expression |
---|---|
raluz2 | ⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluz2 12409 | . . . . . 6 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) | |
2 | 3anass 1097 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛))) | |
3 | 1, 2 | bitri 278 | . . . . 5 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛))) |
4 | 3 | imbi1i 353 | . . . 4 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) → 𝜑) ↔ ((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) → 𝜑)) |
5 | impexp 454 | . . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) → 𝜑))) | |
6 | impexp 454 | . . . . . . 7 ⊢ (((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))) | |
7 | 6 | imbi2i 339 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛) → 𝜑)) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)))) |
8 | 5, 7 | bitri 278 | . . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)))) |
9 | bi2.04 392 | . . . . 5 ⊢ ((𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)))) | |
10 | 8, 9 | bitri 278 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀 ≤ 𝑛)) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)))) |
11 | 4, 10 | bitri 278 | . . 3 ⊢ ((𝑛 ∈ (ℤ≥‘𝑀) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)))) |
12 | 11 | ralbii2 3076 | . 2 ⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑))) |
13 | r19.21v 3088 | . 2 ⊢ (∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀 ≤ 𝑛 → 𝜑)) ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑))) | |
14 | 12, 13 | bitri 278 | 1 ⊢ (∀𝑛 ∈ (ℤ≥‘𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀 ≤ 𝑛 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2112 ∀wral 3051 class class class wbr 5039 ‘cfv 6358 ≤ cle 10833 ℤcz 12141 ℤ≥cuz 12403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-cnex 10750 ax-resscn 10751 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-neg 11030 df-z 12142 df-uz 12404 |
This theorem is referenced by: (None) |
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