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Theorem abrexex 7896
Description: Existence of a class abstraction of existentially restricted sets. See the comment of abrexexg 7894. See also abrexex2 7903. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
abrexex.1 𝐴 ∈ V
Assertion
Ref Expression
abrexex {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem abrexex
StepHypRef Expression
1 abrexex.1 . 2 𝐴 ∈ V
2 abrexexg 7894 . 2 (𝐴 ∈ V → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V)
31, 2ax-mp 5 1 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  {cab 2714  wrex 3074  Vcvv 3446
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  ax-rep 5243
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-v 3448
This theorem is referenced by:  ab2rexex  7913  kmlem10  10096  cshwsexa  14713  shftfval  14956  dvdsrval  20075  cmpsublem  22753  cmpsub  22754  ptrescn  22993  addsproplem2  27285  negsid  27342  satfvsuclem1  33956  fmlasuc0  33981  heibor1lem  36271  pointsetN  38207  eldiophb  41083
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