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Theorem abexex 7979
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 3067 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 abexex.2 . . . . . 6 (𝜑𝑥𝐴)
32pm4.71ri 559 . . . . 5 (𝜑 ↔ (𝑥𝐴𝜑))
43exbii 1842 . . . 4 (∃𝑥𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
51, 4bitr4i 277 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝜑)
65abbii 2797 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑}
7 abexex.1 . . 3 𝐴 ∈ V
8 abexex.3 . . 3 {𝑦𝜑} ∈ V
97, 8abrexex2 7977 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
106, 9eqeltrri 2825 1 {𝑦 ∣ ∃𝑥𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wex 1773  wcel 2098  {cab 2704  wrex 3066  Vcvv 3471
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 2698  ax-rep 5287  ax-sep 5301  ax-un 7744
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 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-v 3473  df-in 3954  df-ss 3964  df-uni 4911  df-iun 5000
This theorem is referenced by:  brdom7disj  10560  brdom6disj  10561
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