Theorem abexssex 7965

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	⊢ {𝑦 ∣ 𝜑} ∈ V

Assertion

Ref	Expression
abexssex	⊢ {𝑦 ∣ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem abexssex

Step	Hyp	Ref	Expression
1		df-rex 3087	. . . 4 ⊢ (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐴 ∧ 𝜑))
2		velpw 4561	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	2	anbi1i 636	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝜑) ↔ (𝑥 ⊆ 𝐴 ∧ 𝜑))
4	3	exbii 1881	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝒫 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑))
5	1, 4	bitri 278	. . 3 ⊢ (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑))
6	5	abbii 2827	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} = {𝑦 ∣ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑)}
7		abrexex2.1	. . . 4 ⊢ 𝐴 ∈ V
8	7	pwex 5341	. . 3 ⊢ 𝒫 𝐴 ∈ V
9		abrexex2.2	. . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V
10	8, 9	abrexex2 7964	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} ∈ V
11	6, 10	eqeltrri 2857	1 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V

This proof depends on syntax axioms:   ∧ wa 401  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∃wrex 3086  Vcvv 3450   ⊆ wss 3898  𝒫 cpw 4556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-pow 5326  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3915  df-pw 4558  df-uni 4867  df-iun 4952

This theorem is used by: (None)