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Theorem abexex 7997
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 3070 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 abexex.2 . . . . . 6 (𝜑𝑥𝐴)
32pm4.71ri 560 . . . . 5 (𝜑 ↔ (𝑥𝐴𝜑))
43exbii 1847 . . . 4 (∃𝑥𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
51, 4bitr4i 278 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝜑)
65abbii 2808 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑}
7 abexex.1 . . 3 𝐴 ∈ V
8 abexex.3 . . 3 {𝑦𝜑} ∈ V
97, 8abrexex2 7995 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
106, 9eqeltrri 2837 1 {𝑦 ∣ ∃𝑥𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1778  wcel 2107  {cab 2713  wrex 3069  Vcvv 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-v 3481  df-ss 3967  df-uni 4907  df-iun 4992
This theorem is referenced by:  brdom7disj  10572  brdom6disj  10573
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