Theorem abexssex 7149

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 2918	. . . 4 ⊢ (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥 ∈ 𝒫 𝐴 ∧ 𝜑))
2		selpw 4165	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	2	anbi1i 731	. . . . 5 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝜑) ↔ (𝑥 ⊆ 𝐴 ∧ 𝜑))
4	3	exbii 1774	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝒫 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑))
5	1, 4	bitri 264	. . 3 ⊢ (∃𝑥 ∈ 𝒫 𝐴𝜑 ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑))
6	5	abbii 2739	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} = {𝑦 ∣ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑)}
7		abrexex2.1	. . . 4 ⊢ 𝐴 ∈ V
8	7	pwex 4848	. . 3 ⊢ 𝒫 𝐴 ∈ V
9		abrexex2.2	. . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V
10	8, 9	abrexex2 7148	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝒫 𝐴𝜑} ∈ V
11	6, 10	eqeltrri 2698	1 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V

Syntax hints:   ∧ wa 384  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by: (None)