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Theorem elintg 4515
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 2718 . . 3 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
21ralbidv 3015 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥))
3 dfint2 4509 . 2 𝐵 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝑥}
42, 3elab2g 3385 1 (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wral 2941   cint 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-int 4508
This theorem is referenced by:  elinti  4517  elrint  4550  onmindif  5853  onmindif2  7054  mremre  16311  toponmre  20945  1stcfb  21296  uffixfr  21774  plycpn  24089  insiga  30328  dfon2lem8  31819  elintabg  38197  trintALTVD  39430  trintALT  39431  elintd  39559  intsaluni  40865  intsal  40866
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