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Theorem elintg 4619
 Description: Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.) (Proof shortened by JJ, 26-Jul-2021.)
Assertion
Ref Expression
elintg (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elintg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2838 . . 3 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
21ralbidv 3135 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥))
3 dfint2 4613 . 2 𝐵 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝑥}
42, 3elab2g 3504 1 (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∩ cint 4611 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-int 4612 This theorem is referenced by:  elinti  4620  elrint  4652  onmindif  5958  onmindif2  7159  mremre  16472  toponmre  21118  1stcfb  21469  uffixfr  21947  plycpn  24264  insiga  30540  dfon2lem8  32031  elintabg  38406  trintALTVD  39638  trintALT  39639  elintd  39766  intsaluni  41064  intsal  41065
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