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Theorem elintg 3650
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 2116 . 2 (𝑦 = 𝐴 → (𝑦 𝐵𝐴 𝐵))
2 eleq1 2116 . . 3 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32ralbidv 2343 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥))
4 vex 2577 . . 3 𝑦 ∈ V
54elint2 3649 . 2 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
61, 3, 5vtoclbg 2631 1 (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  wral 2323   cint 3642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-int 3643
This theorem is referenced by:  elinti  3651  elrint  3682  peano2  4345  pitonn  6981  peano1nnnn  6985  peano2nnnn  6986  1nn  8000  peano2nn  8001
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