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Theorem abexssex 7996
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 3070 . . . 4 (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐴𝜑))
2 velpw 4604 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32anbi1i 624 . . . . 5 ((𝑥 ∈ 𝒫 𝐴𝜑) ↔ (𝑥𝐴𝜑))
43exbii 1847 . . . 4 (∃𝑥(𝑥 ∈ 𝒫 𝐴𝜑) ↔ ∃𝑥(𝑥𝐴𝜑))
51, 4bitri 275 . . 3 (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
65abbii 2808 . 2 {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)}
7 abrexex2.1 . . . 4 𝐴 ∈ V
87pwex 5379 . . 3 𝒫 𝐴 ∈ V
9 abrexex2.2 . . 3 {𝑦𝜑} ∈ V
108, 9abrexex2 7995 . 2 {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} ∈ V
116, 10eqeltrri 2837 1 {𝑦 ∣ ∃𝑥(𝑥𝐴𝜑)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 395  wex 1778  wcel 2107  {cab 2713  wrex 3069  Vcvv 3479  wss 3950  𝒫 cpw 4599
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-pow 5364  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-pw 4601  df-uni 4907  df-iun 4992
This theorem is referenced by: (None)
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