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Theorem abrexex2 7903
Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 7896. (Contributed by NM, 12-Sep-2004.)
Hypotheses
Ref Expression
abrexex2.1 𝐴 ∈ V
abrexex2.2 {𝑦𝜑} ∈ V
Assertion
Ref Expression
abrexex2 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem abrexex2
StepHypRef Expression
1 abrexex2.1 . 2 𝐴 ∈ V
2 abrexex2.2 . . 3 {𝑦𝜑} ∈ V
32rgenw 3069 . 2 𝑥𝐴 {𝑦𝜑} ∈ V
4 abrexex2g 7898 . 2 ((𝐴 ∈ V ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ V) → {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V)
51, 3, 4mp2an 691 1 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  {cab 2714  wral 3065  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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-v 3448  df-in 3918  df-ss 3928  df-uni 4867  df-iun 4957
This theorem is referenced by:  abexssex  7904  abexex  7905  oprabrexex2  7912  ab2rexex  7913  ab2rexex2  7914
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