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Theorem abrexex2 7974
Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 7967. (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 3054 . 2 𝑥𝐴 {𝑦𝜑} ∈ V
4 abrexex2g 7969 . 2 ((𝐴 ∈ V ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ V) → {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V)
51, 3, 4mp2an 690 1 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  {cab 2702  wral 3050  wrex 3059  Vcvv 3461
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 2696  ax-rep 5286  ax-sep 5300  ax-un 7741
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 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-v 3463  df-ss 3961  df-uni 4910  df-iun 4999
This theorem is referenced by:  abexssex  7975  abexex  7976  oprabrexex2  7983  ab2rexex  7984  ab2rexex2  7985
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