Theorem ab2rexex2 7699

Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 7688. (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 7688	. 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝜑} ∈ V
5	1, 4	abrexex2 7688	1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V

Syntax hints:   ∈ wcel 2113  {cab 2716  ∃wrex 3054  Vcvv 3397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341

This theorem is referenced by:  brdom7disj  10024  brdom6disj  10025  lineset  37364