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Theorem elintg 4959
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 2822 . . 3 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
21ralbidv 3178 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥))
3 dfint2 4953 . 2 𝐵 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝑥}
42, 3elab2g 3671 1 (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  wral 3062   cint 4951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-int 4952
This theorem is referenced by:  elinti  4960  elintabg  4962  elrint  4996  onmindif  6457  onmindif2  7795  mremre  17548  toponmre  22597  1stcfb  22949  uffixfr  23427  plycpn  25802  insiga  33135  dfon2lem8  34762  trintALTVD  43641  trintALT  43642  elintd  43763  intsaluni  45045  intsal  45046
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