MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ab2rexex2 Structured version   Visualization version   GIF version

Theorem ab2rexex2 7966
Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 7955. (Contributed by NM, 20-Sep-2011.)
Hypotheses
Ref Expression
ab2rexex2.1 𝐴 ∈ V
ab2rexex2.2 𝐵 ∈ V
ab2rexex2.3 {𝑧𝜑} ∈ V
Assertion
Ref Expression
ab2rexex2 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 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
StepHypRef Expression
1 ab2rexex2.1 . 2 𝐴 ∈ V
2 ab2rexex2.2 . . 3 𝐵 ∈ V
3 ab2rexex2.3 . . 3 {𝑧𝜑} ∈ V
42, 3abrexex2 7955 . 2 {𝑧 ∣ ∃𝑦𝐵 𝜑} ∈ V
51, 4abrexex2 7955 1 {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝜑} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  {cab 2703  wrex 3064  Vcvv 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-v 3470  df-in 3950  df-ss 3960  df-uni 4903  df-iun 4992
This theorem is referenced by:  brdom7disj  10528  brdom6disj  10529  lineset  39122
  Copyright terms: Public domain W3C validator