Theorem ab2rexex2 7965

Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 7954. (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 7954	. 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝜑} ∈ V
5	1, 4	abrexex2 7954	1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V

Syntax hints:   ∈ wcel 2145  {cab 2743  ∃wrex 3089  Vcvv 3457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-v 3459  df-ss 3924  df-uni 4869  df-iun 4954

This theorem is referenced by:  brdom7disj  10503  brdom6disj  10504  lineset  40374