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Theorem elintg 3673
Description: Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
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 2147 . 2 (𝑦 = 𝐴 → (𝑦 𝐵𝐴 𝐵))
2 eleq1 2147 . . 3 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32ralbidv 2376 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥))
4 vex 2617 . . 3 𝑦 ∈ V
54elint2 3672 . 2 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
61, 3, 5vtoclbg 2672 1 (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1287  wcel 1436  wral 2355   cint 3665
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2616  df-int 3666
This theorem is referenced by:  elinti  3674  elrint  3705  peano2  4376  pitonn  7306  peano1nnnn  7310  peano2nnnn  7311  1nn  8345  peano2nn  8346
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