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Theorem elintg 3779
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 2202 . 2 (𝑦 = 𝐴 → (𝑦 𝐵𝐴 𝐵))
2 eleq1 2202 . . 3 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32ralbidv 2437 . 2 (𝑦 = 𝐴 → (∀𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥))
4 vex 2689 . . 3 𝑦 ∈ V
54elint2 3778 . 2 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
61, 3, 5vtoclbg 2747 1 (𝐴𝑉 → (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wcel 1480  wral 2416   cint 3771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-int 3772
This theorem is referenced by:  elinti  3780  elrint  3811  peano2  4509  pitonn  7670  peano1nnnn  7674  peano2nnnn  7675  1nn  8745  peano2nn  8746
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