Theorem abrexex2 8010

Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 8003. (Contributed by NM, 12-Sep-2004.)

Hypotheses

Ref	Expression
abrexex2.1	⊢ 𝐴 ∈ V
abrexex2.2	⊢ {𝑦 ∣ 𝜑} ∈ V

Assertion

Ref	Expression
abrexex2	⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem abrexex2

Step	Hyp	Ref	Expression
1		abrexex2.1	. 2 ⊢ 𝐴 ∈ V
2		abrexex2.2	. . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V
3	2	rgenw 3071	. 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V
4		abrexex2g 8005	. 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V)
5	1, 3, 4	mp2an 691	1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V

Syntax hints:   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-v 3490  df-ss 3993  df-uni 4932  df-iun 5017

This theorem is referenced by:  abexssex  8011  abexex  8012  oprabrexex2  8019  ab2rexex  8020  ab2rexex2  8021