Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  abrexex2 Structured version   Visualization version   GIF version

Theorem abrexex2 7662
 Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 7655. (Contributed by NM, 12-Sep-2004.)
Hypotheses
Ref Expression
abrexex2.1 𝐴 ∈ V
abrexex2.2 {𝑦𝜑} ∈ V
Assertion
Ref Expression
abrexex2 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem abrexex2
StepHypRef Expression
1 abrexex2.1 . 2 𝐴 ∈ V
2 abrexex2.2 . . 3 {𝑦𝜑} ∈ V
32rgenw 3145 . 2 𝑥𝐴 {𝑦𝜑} ∈ V
4 abrexex2g 7657 . 2 ((𝐴 ∈ V ∧ ∀𝑥𝐴 {𝑦𝜑} ∈ V) → {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V)
51, 3, 4mp2an 691 1 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2115  {cab 2802  ∀wral 3133  ∃wrex 3134  Vcvv 3480 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pr 5318  ax-un 7452 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352 This theorem is referenced by:  abexssex  7663  abexex  7664  oprabrexex2  7671  ab2rexex  7672  ab2rexex2  7673
 Copyright terms: Public domain W3C validator