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Theorem elintg 3614
 Description: Membership in class intersection, with the sethood requirement expressed as an antecedent. (Contributed by NM, 20-Nov-2003.)
Assertion
Ref Expression
elintg (A 𝑉 → (A Bx B A x))
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 2097 . 2 (y = A → (y BA B))
2 eleq1 2097 . . 3 (y = A → (y xA x))
32ralbidv 2320 . 2 (y = A → (x B y xx B A x))
4 vex 2554 . . 3 y V
54elint2 3613 . 2 (y Bx B y x)
61, 3, 5vtoclbg 2608 1 (A 𝑉 → (A Bx B A x))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  ∩ cint 3606 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-int 3607 This theorem is referenced by:  elinti  3615  elrint  3646  peano2  4261  pitonn  6744  1nn  7706  peano2nn  7707
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