Theorem abrexex2 7923

Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 7916. (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 3056	. 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V
4		abrexex2g 7918	. 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V)
5	1, 3, 4	mp2an 693	1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V

Syntax hints:   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  Vcvv 3442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-v 3444  df-ss 3920  df-uni 4866  df-iun 4950

This theorem is referenced by:  abexssex  7924  abexex  7925  oprabrexex2  7932  ab2rexex  7933  ab2rexex2  7934