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Theorem ab2rexex 8004
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 7987. (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 7987 . 2 {𝑧 ∣ ∃𝑦𝐵 𝑧 = 𝐶} ∈ V
41, 3abrexex2 7994 1 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-uni 4908  df-iun 4993
This theorem is referenced by:  plyval  26232  precsexlem4  28234  precsexlem5  28235  onmulscl  28287  zs12ex  28422  pstmfval  33895  pstmxmet  33896
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