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Theorem abexex 7975
Description: A condition where a class abstraction continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
Hypotheses
Ref Expression
abexex.1 𝐴 ∈ V
abexex.2 (𝜑𝑥𝐴)
abexex.3 {𝑦𝜑} ∈ V
Assertion
Ref Expression
abexex {𝑦 ∣ ∃𝑥𝜑} ∈ V
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem abexex
StepHypRef Expression
1 df-rex 3068 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 abexex.2 . . . . . 6 (𝜑𝑥𝐴)
32pm4.71ri 560 . . . . 5 (𝜑 ↔ (𝑥𝐴𝜑))
43exbii 1843 . . . 4 (∃𝑥𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
51, 4bitr4i 278 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝜑)
65abbii 2798 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑}
7 abexex.1 . . 3 𝐴 ∈ V
8 abexex.3 . . 3 {𝑦𝜑} ∈ V
97, 8abrexex2 7973 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
106, 9eqeltrri 2826 1 {𝑦 ∣ ∃𝑥𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1774  wcel 2099  {cab 2705  wrex 3067  Vcvv 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-v 3473  df-in 3954  df-ss 3964  df-uni 4909  df-iun 4998
This theorem is referenced by:  brdom7disj  10555  brdom6disj  10556
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