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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 2827 . . 3 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
21ralbidv 3176 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥))
3 dfint2 4953 . 2 𝐵 = {𝑦 ∣ ∀𝑥𝐵 𝑦𝑥}
42, 3elab2g 3683 1 (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  wral 3059   cint 4951
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-int 4952
This theorem is referenced by:  elinti  4960  elintabg  4962  elrint  4994  onmindif  6478  onmindif2  7827  mremre  17649  toponmre  23117  1stcfb  23469  uffixfr  23947  plycpn  26346  insiga  34118  dfon2lem8  35772  trintALTVD  44878  trintALT  44879  elintd  45014  intsaluni  46285  intsal  46286
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