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Theorem abexssex 7904
Description: Existence of a class abstraction with an existentially quantified expression. Both 𝑥 and 𝑦 can be free in 𝜑. (Contributed by NM, 29-Jul-2006.)
Hypotheses
Ref Expression
abrexex2.1 𝐴 ∈ V
abrexex2.2 {𝑦𝜑} ∈ V
Assertion
Ref Expression
abexssex {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)} ∈ V
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem abexssex
StepHypRef Expression
1 df-rex 3071 . . . 4 (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐴𝜑))
2 velpw 4566 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32anbi1i 625 . . . . 5 ((𝑥 ∈ 𝒫 𝐴𝜑) ↔ (𝑥𝐴𝜑))
43exbii 1851 . . . 4 (∃𝑥(𝑥 ∈ 𝒫 𝐴𝜑) ↔ ∃𝑥(𝑥𝐴𝜑))
51, 4bitri 275 . . 3 (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
65abbii 2803 . 2 {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)}
7 abrexex2.1 . . . 4 𝐴 ∈ V
87pwex 5336 . . 3 𝒫 𝐴 ∈ V
9 abrexex2.2 . . 3 {𝑦𝜑} ∈ V
108, 9abrexex2 7903 . 2 {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} ∈ V
116, 10eqeltrri 2831 1 {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 397  wex 1782  wcel 2107  {cab 2710  wrex 3070  Vcvv 3444  wss 3911  𝒫 cpw 4561
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 2704  ax-rep 5243  ax-sep 5257  ax-pow 5321  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 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-v 3446  df-in 3918  df-ss 3928  df-pw 4563  df-uni 4867  df-iun 4957
This theorem is referenced by: (None)
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