Theorem abexssex 7951

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 3088	. . . 4 ⊢ (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐴 ∧ 𝜑))
2		velpw 4561	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	2	anbi1i 633	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝜑) ↔ (𝑥 ⊆ 𝐴 ∧ 𝜑))
4	3	exbii 1869	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝒫 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑))
5	1, 4	bitri 277	. . 3 ⊢ (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑))
6	5	abbii 2830	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} = {𝑦 ∣ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑)}
7		abrexex2.1	. . . 4 ⊢ 𝐴 ∈ V
8	7	pwex 5338	. . 3 ⊢ 𝒫 𝐴 ∈ V
9		abrexex2.2	. . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V
10	8, 9	abrexex2 7950	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} ∈ V
11	6, 10	eqeltrri 2860	1 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V

Syntax hints:   ∧ wa 399  ∃wex 1800   ∈ wcel 2143  {cab 2741  ∃wrex 3087  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-pow 5323  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1564  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3078  df-rex 3088  df-v 3457  df-ss 3922  df-pw 4558  df-uni 4867  df-iun 4952

This theorem is referenced by: (None)