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Theorem ralrab 3683
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 3678 . . . 4 (𝑥 ∈ {𝑦𝐴𝜑} ↔ (𝑥𝐴𝜓))
32imbi1i 352 . . 3 ((𝑥 ∈ {𝑦𝐴𝜑} → 𝜒) ↔ ((𝑥𝐴𝜓) → 𝜒))
4 impexp 453 . . 3 (((𝑥𝐴𝜓) → 𝜒) ↔ (𝑥𝐴 → (𝜓𝜒)))
53, 4bitri 277 . 2 ((𝑥 ∈ {𝑦𝐴𝜑} → 𝜒) ↔ (𝑥𝐴 → (𝜓𝜒)))
65ralbii2 3161 1 (∀𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∀𝑥𝐴 (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2108  wral 3136  {crab 3140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rab 3145  df-v 3495
This theorem is referenced by:  frminex  5528  wereu2  5545  weniso  7099  zmin  12336  prmreclem1  16244  lublecllem  17590  mhmeql  17982  ghmeql  18373  pgpfac1lem5  19193  lmhmeql  19819  islindf4  20974  1stcfb  22045  fbssfi  22437  filssufilg  22511  txflf  22606  ptcmplem3  22654  symgtgp  22706  tgpconncompeqg  22712  cnllycmp  23552  ovolgelb  24073  dyadmax  24191  lhop1  24603  radcnvlt1  24998  frpomin  33071  noextenddif  33168  conway  33257  poimirlem4  34888  poimirlem32  34916  ismblfin  34925  igenval2  35336  glbconN  36505  mgmhmeql  44060
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