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Theorem abexssex 7984
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 3061 . . . 4 (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐴𝜑))
2 velpw 4612 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32anbi1i 622 . . . . 5 ((𝑥 ∈ 𝒫 𝐴𝜑) ↔ (𝑥𝐴𝜑))
43exbii 1843 . . . 4 (∃𝑥(𝑥 ∈ 𝒫 𝐴𝜑) ↔ ∃𝑥(𝑥𝐴𝜑))
51, 4bitri 274 . . 3 (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
65abbii 2796 . 2 {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)}
7 abrexex2.1 . . . 4 𝐴 ∈ V
87pwex 5384 . . 3 𝒫 𝐴 ∈ V
9 abrexex2.2 . . 3 {𝑦𝜑} ∈ V
108, 9abrexex2 7983 . 2 {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} ∈ V
116, 10eqeltrri 2823 1 {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 394  wex 1774  wcel 2099  {cab 2703  wrex 3060  Vcvv 3462  wss 3947  𝒫 cpw 4607
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 2697  ax-rep 5290  ax-sep 5304  ax-pow 5369  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-v 3464  df-ss 3964  df-pw 4609  df-uni 4914  df-iun 5003
This theorem is referenced by: (None)
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