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Theorem ralrab 3689
Description: Universal quantification over a restricted class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypothesis
Ref Expression
ralab.1 (𝑦 = 𝑥 → (𝜑𝜓))
Assertion
Ref Expression
ralrab (∀𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∀𝑥𝐴 (𝜓𝜒))
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem ralrab
StepHypRef Expression
1 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
21elrab 3683 . . . 4 (𝑥 ∈ {𝑦𝐴𝜑} ↔ (𝑥𝐴𝜓))
32imbi1i 349 . . 3 ((𝑥 ∈ {𝑦𝐴𝜑} → 𝜒) ↔ ((𝑥𝐴𝜓) → 𝜒))
4 impexp 451 . . 3 (((𝑥𝐴𝜓) → 𝜒) ↔ (𝑥𝐴 → (𝜓𝜒)))
53, 4bitri 274 . 2 ((𝑥 ∈ {𝑦𝐴𝜑} → 𝜒) ↔ (𝑥𝐴 → (𝜓𝜒)))
65ralbii2 3089 1 (∀𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∀𝑥𝐴 (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  wral 3061  {crab 3432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476
This theorem is referenced by:  frminex  5656  wereu2  5673  frpomin  6341  weniso  7353  zmin  12932  prmreclem1  16853  lublecllem  18317  mgmhmeql  18641  mhmeql  18743  ghmeql  19153  pgpfac1lem5  19990  lmhmeql  20810  islindf4  21612  1stcfb  23169  fbssfi  23561  filssufilg  23635  txflf  23730  ptcmplem3  23778  symgtgp  23830  tgpconncompeqg  23836  cnllycmp  24696  ovolgelb  25221  dyadmax  25339  lhop1  25755  radcnvlt1  26154  noextenddif  27395  conway  27525  madebdaylemlrcut  27618  poimirlem4  36795  poimirlem32  36823  ismblfin  36832  igenval2  37237  glbconN  38550  glbconNOLD  38551  nadd2rabtr  42436  intubeu  47697  unilbeu  47698
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