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Theorem raluz 9001
Description: Restricted universal quantification in an upper set of integers. (Contributed by NM, 9-Sep-2005.)
Assertion
Ref Expression
raluz (𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Distinct variable group:   𝑛,𝑀
Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem raluz
StepHypRef Expression
1 eluz1 8958 . . . 4 (𝑀 ∈ ℤ → (𝑛 ∈ (ℤ𝑀) ↔ (𝑛 ∈ ℤ ∧ 𝑀𝑛)))
21imbi1d 229 . . 3 (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ ((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑)))
3 impexp 259 . . 3 (((𝑛 ∈ ℤ ∧ 𝑀𝑛) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀𝑛𝜑)))
42, 3syl6bb 194 . 2 (𝑀 ∈ ℤ → ((𝑛 ∈ (ℤ𝑀) → 𝜑) ↔ (𝑛 ∈ ℤ → (𝑀𝑛𝜑))))
54ralbidv2 2378 1 (𝑀 ∈ ℤ → (∀𝑛 ∈ (ℤ𝑀)𝜑 ↔ ∀𝑛 ∈ ℤ (𝑀𝑛𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1436  wral 2355   class class class wbr 3822  cfv 4983  cle 7470  cz 8686  cuz 8954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-cnex 7383  ax-resscn 7384
This theorem depends on definitions:  df-bi 115  df-3or 923  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-mpt 3878  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-iota 4948  df-fun 4985  df-fv 4991  df-ov 5618  df-neg 7603  df-z 8687  df-uz 8955
This theorem is referenced by: (None)
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