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Theorem elintg 4958
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 2821 . . 3 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
21ralbidv 3177 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥))
3 dfint2 4952 . 2 𝐵 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝑥}
42, 3elab2g 3670 1 (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  wral 3061   cint 4950
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-int 4951
This theorem is referenced by:  elinti  4959  elintabg  4961  elrint  4995  onmindif  6456  onmindif2  7797  mremre  17552  toponmre  22817  1stcfb  23169  uffixfr  23647  plycpn  26026  insiga  33421  dfon2lem8  35054  trintALTVD  43943  trintALT  43944  elintd  44065  intsaluni  45344  intsal  45345
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