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Theorem ralrab 3656
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 3650 . . . 4 (𝑥 ∈ {𝑦𝐴𝜑} ↔ (𝑥𝐴𝜓))
32imbi1i 350 . . 3 ((𝑥 ∈ {𝑦𝐴𝜑} → 𝜒) ↔ ((𝑥𝐴𝜓) → 𝜒))
4 impexp 452 . . 3 (((𝑥𝐴𝜓) → 𝜒) ↔ (𝑥𝐴 → (𝜓𝜒)))
53, 4bitri 275 . 2 ((𝑥 ∈ {𝑦𝐴𝜑} → 𝜒) ↔ (𝑥𝐴 → (𝜓𝜒)))
65ralbii2 3093 1 (∀𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∀𝑥𝐴 (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  wral 3065  {crab 3410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450
This theorem is referenced by:  frminex  5618  wereu2  5635  frpomin  6299  weniso  7304  zmin  12876  prmreclem1  16795  lublecllem  18256  mhmeql  18643  ghmeql  19038  pgpfac1lem5  19865  lmhmeql  20532  islindf4  21260  1stcfb  22812  fbssfi  23204  filssufilg  23278  txflf  23373  ptcmplem3  23421  symgtgp  23473  tgpconncompeqg  23479  cnllycmp  24335  ovolgelb  24860  dyadmax  24978  lhop1  25394  radcnvlt1  25793  noextenddif  27032  conway  27160  madebdaylemlrcut  27250  poimirlem4  36111  poimirlem32  36139  ismblfin  36148  igenval2  36554  glbconN  37868  glbconNOLD  37869  nadd2rabtr  41729  mgmhmeql  46171  intubeu  47083  unilbeu  47084
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