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Theorem rexrab 3693
Description: Existential quantification over a class abstraction. (Contributed by Jeff Madsen, 17-Jun-2011.) (Revised by Mario Carneiro, 3-Sep-2015.)
Hypothesis
Ref Expression
ralab.1 (𝑦 = 𝑥 → (𝜑𝜓))
Assertion
Ref Expression
rexrab (∃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 rexrab
StepHypRef Expression
1 ralab.1 . . . . 5 (𝑦 = 𝑥 → (𝜑𝜓))
21elrab 3684 . . . 4 (𝑥 ∈ {𝑦𝐴𝜑} ↔ (𝑥𝐴𝜓))
32anbi1i 625 . . 3 ((𝑥 ∈ {𝑦𝐴𝜑} ∧ 𝜒) ↔ ((𝑥𝐴𝜓) ∧ 𝜒))
4 anass 470 . . 3 (((𝑥𝐴𝜓) ∧ 𝜒) ↔ (𝑥𝐴 ∧ (𝜓𝜒)))
53, 4bitri 275 . 2 ((𝑥 ∈ {𝑦𝐴𝜑} ∧ 𝜒) ↔ (𝑥𝐴 ∧ (𝜓𝜒)))
65rexbii2 3091 1 (∃𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∃𝑥𝐴 (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  wrex 3071  {crab 3433
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 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-rab 3434  df-v 3477
This theorem is referenced by:  wereu2  5674  frpomin  6342  wdom2d  9575  enfin2i  10316  infm3  12173  pmtrfrn  19326  pgpssslw  19482  ellspd  21357  1stcfb  22949  xkobval  23090  xkococn  23164  imasdsf1olem  23879  eqscut2  27307  scutun12  27311  cuteq0  27333  rusgrnumwwlks  29228  cvmliftlem15  34289  wsuclem  34797  poimirlem4  36492  poimirlem26  36514  poimirlem27  36515  infdesc  41385  rexrabdioph  41532  hbtlem6  41871
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