Theorem ab2rexex2 7961

Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 7950. (Contributed by NM, 20-Sep-2011.)

Hypotheses

Ref	Expression
ab2rexex2.1	⊢ 𝐴 ∈ V
ab2rexex2.2	⊢ 𝐵 ∈ V
ab2rexex2.3	⊢ {𝑧 ∣ 𝜑} ∈ V

Assertion

Ref	Expression
ab2rexex2	⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V

Distinct variable groups:   𝑥,𝑧,𝐴   𝑦,𝑧,𝐵

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐴(𝑦)   𝐵(𝑥)

Proof of Theorem ab2rexex2

Step	Hyp	Ref	Expression
1		ab2rexex2.1	. 2 ⊢ 𝐴 ∈ V
2		ab2rexex2.2	. . 3 ⊢ 𝐵 ∈ V
3		ab2rexex2.3	. . 3 ⊢ {𝑧 ∣ 𝜑} ∈ V
4	2, 3	abrexex2 7950	. 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝜑} ∈ V
5	1, 4	abrexex2 7950	1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V

Syntax hints:   ∈ wcel 2098  {cab 2701  ∃wrex 3062  Vcvv 3466

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-v 3468  df-in 3948  df-ss 3958  df-uni 4901  df-iun 4990

This theorem is referenced by:  brdom7disj  10523  brdom6disj  10524  lineset  39103