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

Theorem abexex 7666
 Description: A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
Hypotheses
Ref Expression
abexex.1 𝐴 ∈ V
abexex.2 (𝜑𝑥𝐴)
abexex.3 {𝑦𝜑} ∈ V
Assertion
Ref Expression
abexex {𝑦 ∣ ∃𝑥𝜑} ∈ V
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem abexex
StepHypRef Expression
1 df-rex 3144 . . . 4 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 abexex.2 . . . . . 6 (𝜑𝑥𝐴)
32pm4.71ri 563 . . . . 5 (𝜑 ↔ (𝑥𝐴𝜑))
43exbii 1844 . . . 4 (∃𝑥𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
51, 4bitr4i 280 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥𝜑)
65abbii 2886 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} = {𝑦 ∣ ∃𝑥𝜑}
7 abexex.1 . . 3 𝐴 ∈ V
8 abexex.3 . . 3 {𝑦𝜑} ∈ V
97, 8abrexex2 7664 . 2 {𝑦 ∣ ∃𝑥𝐴 𝜑} ∈ V
106, 9eqeltrri 2910 1 {𝑦 ∣ ∃𝑥𝜑} ∈ V
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1776   ∈ wcel 2110  {cab 2799  ∃wrex 3139  Vcvv 3495 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358 This theorem is referenced by:  brdom7disj  9947  brdom6disj  9948
 Copyright terms: Public domain W3C validator