Theorem ab2rexex2 8026

Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 8015. (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 8015	. 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝜑} ∈ V
5	1, 4	abrexex2 8015	1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V

This proof depends on syntax axioms:   ∈ wcel 2147  {cab 2741  ∃wrex 3090  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-v 3456  df-ss 3901  df-uni 4873  df-iun 4958

This theorem is used by:  brdom7disj  10649  brdom6disj  10650  lineset  41061