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Theorem ab2rexex 7666
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 7649. (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 7649 . 2 {𝑧 ∣ ∃𝑦𝐵 𝑧 = 𝐶} ∈ V
41, 3abrexex2 7656 1 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶} ∈ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2112  {cab 2779  wrex 3110  Vcvv 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336
This theorem is referenced by:  plyval  24794  pstmfval  31253  pstmxmet  31254
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