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Theorem raluz2 9430
 Description: Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
Assertion
Ref Expression
raluz2 (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Distinct variable group:   𝑛,𝑀
Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem raluz2
StepHypRef Expression
1 eluz2 9385 . . . . . 6 (𝑛 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀𝑛))
2 3anass 967 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ ∧ 𝑀𝑛) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
31, 2bitri 183 . . . . 5 (𝑛 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
43imbi1i 237 . . . 4 ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ ((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑))
5 impexp 261 . . . . . 6 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑)))
6 impexp 261 . . . . . . 7 (((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀𝑛𝜑)))
76imbi2i 225 . . . . . 6 ((𝑀 ∈ ℤ → ((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑)) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
85, 7bitri 183 . . . . 5 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
9 bi2.04 247 . . . . 5 ((𝑀 ∈ ℤ → (𝑛 ∈ ℤ → (𝑀𝑛𝜑))) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
108, 9bitri 183 . . . 4 (((𝑀 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑀𝑛)) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
114, 10bitri 183 . . 3 ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀𝑛𝜑))))
1211ralbii2 2450 . 2 (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀𝑛𝜑)))
13 r19.21v 2514 . 2 (∀𝑛 ∈ ℤ (𝑀 ∈ ℤ → (𝑀𝑛𝜑)) ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
1412, 13bitri 183 1 (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ (𝑀 ∈ ℤ → ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   ∈ wcel 1481  ∀wral 2418   class class class wbr 3939  ‘cfv 5135   ≤ cle 7854  ℤcz 9107  ℤ≥cuz 9379 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142  ax-cnex 7764  ax-resscn 7765 This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1738  df-eu 2004  df-mo 2005  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-rab 2427  df-v 2693  df-sbc 2916  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-br 3940  df-opab 4000  df-mpt 4001  df-id 4226  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-res 4563  df-ima 4564  df-iota 5100  df-fun 5137  df-fn 5138  df-f 5139  df-fv 5143  df-ov 5789  df-neg 7989  df-z 9108  df-uz 9380 This theorem is referenced by: (None)
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