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Theorem elint 3865
Description: Membership in class intersection. (Contributed by NM, 21-May-1994.)
Hypothesis
Ref Expression
elint.1 𝐴 ∈ V
Assertion
Ref Expression
elint (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elint
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elint.1 . 2 𝐴 ∈ V
2 eleq1 2252 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32imbi2d 230 . . 3 (𝑦 = 𝐴 → ((𝑥𝐵𝑦𝑥) ↔ (𝑥𝐵𝐴𝑥)))
43albidv 1835 . 2 (𝑦 = 𝐴 → (∀𝑥(𝑥𝐵𝑦𝑥) ↔ ∀𝑥(𝑥𝐵𝐴𝑥)))
5 df-int 3860 . 2 𝐵 = {𝑦 ∣ ∀𝑥(𝑥𝐵𝑦𝑥)}
61, 4, 5elab2 2900 1 (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1362   = wceq 1364  wcel 2160  Vcvv 2752   cint 3859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-int 3860
This theorem is referenced by:  elint2  3866  elintab  3870  intss1  3874  intss  3880  intun  3890  intpr  3891  peano1  4611
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