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Theorem ab2rexex 7111
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 7094. (Contributed by NM, 20-Sep-2011.)
Hypotheses
Ref Expression
ab2rexex.1 𝐴 ∈ V
ab2rexex.2 𝐵 ∈ V
Assertion
Ref Expression
ab2rexex {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 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
StepHypRef Expression
1 ab2rexex.1 . 2 𝐴 ∈ V
2 ab2rexex.2 . . 3 𝐵 ∈ V
32abrexex 7094 . 2 {𝑧 ∣ ∃𝑦𝐵 𝑧 = 𝐶} ∈ V
41, 3abrexex2 7101 1 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  Vcvv 3189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860
This theorem is referenced by:  plyval  23870  pstmfval  29745  pstmxmet  29746
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