Theorem ab2rexex2 7959

Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 7948. (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 7948	. 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝜑} ∈ V
5	1, 4	abrexex2 7948	1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V

Syntax hints:   ∈ wcel 2109  {cab 2707  ∃wrex 3053  Vcvv 3447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-v 3449  df-ss 3931  df-uni 4872  df-iun 4957

This theorem is referenced by:  brdom7disj  10484  brdom6disj  10485  lineset  39732