Theorem ab2rexex 7964

Description: Existence of a class abstraction of existentially restricted sets. Variables 𝑥 and 𝑦 are normally free-variable parameters in the class expression substituted for 𝐶, which can be thought of as 𝐶(𝑥, 𝑦). See comments for abrexex 7947. (Contributed by NM, 20-Sep-2011.)

Hypotheses

Ref	Expression
ab2rexex.1	⊢ 𝐴 ∈ V
ab2rexex.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
ab2rexex	⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V

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

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem ab2rexex

Step	Hyp	Ref	Expression
1		ab2rexex.1	. 2 ⊢ 𝐴 ∈ V
2		ab2rexex.2	. . 3 ⊢ 𝐵 ∈ V
3	2	abrexex 7947	. 2 ⊢ {𝑧 ∣ ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V
4	1, 3	abrexex2 7954	1 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V

Syntax hints:   = wceq 1563   ∈ 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:  plyval  26311  precsexlem4  28361  precsexlem5  28362  onmulscl  28429  z12sex  28625  pstmfval  34203  pstmxmet  34204