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Theorem ab2rexex 7989
Description: Existence of a class abstraction of existentially restricted sets. Variables 𝑥 and 𝑦 are normally free-variable parameters in the class expression substituted for 𝐶, which can be thought of as 𝐶(𝑥, 𝑦). See comments for abrexex 7972. (Contributed by NM, 20-Sep-2011.)
Hypotheses
Ref Expression
ab2rexex.1 𝐴 ∈ V
ab2rexex.2 𝐵 ∈ V
Assertion
Ref Expression
ab2rexex {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
Distinct variable groups:   𝑥,𝑧,𝐴   𝑦,𝑧,𝐵   𝑧,𝐶
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem ab2rexex
StepHypRef Expression
1 ab2rexex.1 . 2 𝐴 ∈ V
2 ab2rexex.2 . . 3 𝐵 ∈ V
32abrexex 7972 . 2 {𝑧 ∣ ∃𝑦𝐵 𝑧 = 𝐶} ∈ V
41, 3abrexex2 7979 1 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  {cab 2705  wrex 3067  Vcvv 3473
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-v 3475  df-in 3956  df-ss 3966  df-uni 4913  df-iun 5002
This theorem is referenced by:  plyval  26147  precsexlem4  28128  precsexlem5  28129  pstmfval  33530  pstmxmet  33531
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