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Theorem rexrab 3684
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 3677 . . . 4 (𝑥 ∈ {𝑦𝐴𝜑} ↔ (𝑥𝐴𝜓))
32anbi1i 623 . . 3 ((𝑥 ∈ {𝑦𝐴𝜑} ∧ 𝜒) ↔ ((𝑥𝐴𝜓) ∧ 𝜒))
4 anass 469 . . 3 (((𝑥𝐴𝜓) ∧ 𝜒) ↔ (𝑥𝐴 ∧ (𝜓𝜒)))
53, 4bitri 276 . 2 ((𝑥 ∈ {𝑦𝐴𝜑} ∧ 𝜒) ↔ (𝑥𝐴 ∧ (𝜓𝜒)))
65rexbii2 3242 1 (∃𝑥 ∈ {𝑦𝐴𝜑}𝜒 ↔ ∃𝑥𝐴 (𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wcel 2105  wrex 3136  {crab 3139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494
This theorem is referenced by:  wereu2  5545  wdom2d  9032  enfin2i  9731  infm3  11588  pmtrfrn  18515  pgpssslw  18668  ellspd  20874  1stcfb  21981  xkobval  22122  xkococn  22196  imasdsf1olem  22910  rusgrnumwwlks  27680  cvmliftlem15  32442  frpomin  32975  wsuclem  33009  scutun12  33168  poimirlem4  34777  poimirlem26  34799  poimirlem27  34800  rexrabdioph  39269  hbtlem6  39607
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