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Theorem abexssex 7956
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 3065 . . . 4 (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐴𝜑))
2 velpw 4602 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32anbi1i 623 . . . . 5 ((𝑥 ∈ 𝒫 𝐴𝜑) ↔ (𝑥𝐴𝜑))
43exbii 1842 . . . 4 (∃𝑥(𝑥 ∈ 𝒫 𝐴𝜑) ↔ ∃𝑥(𝑥𝐴𝜑))
51, 4bitri 275 . . 3 (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
65abbii 2796 . 2 {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)}
7 abrexex2.1 . . . 4 𝐴 ∈ V
87pwex 5371 . . 3 𝒫 𝐴 ∈ V
9 abrexex2.2 . . 3 {𝑦𝜑} ∈ V
108, 9abrexex2 7955 . 2 {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} ∈ V
116, 10eqeltrri 2824 1 {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 395  wex 1773  wcel 2098  {cab 2703  wrex 3064  Vcvv 3468  wss 3943  𝒫 cpw 4597
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-pow 5356  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599  df-uni 4903  df-iun 4992
This theorem is referenced by: (None)
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