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Theorem elint 3649
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 2116 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32imbi2d 223 . . 3 (𝑦 = 𝐴 → ((𝑥𝐵𝑦𝑥) ↔ (𝑥𝐵𝐴𝑥)))
43albidv 1721 . 2 (𝑦 = 𝐴 → (∀𝑥(𝑥𝐵𝑦𝑥) ↔ ∀𝑥(𝑥𝐵𝐴𝑥)))
5 df-int 3644 . 2 𝐵 = {𝑦 ∣ ∀𝑥(𝑥𝐵𝑦𝑥)}
61, 4, 5elab2 2713 1 (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257   = wceq 1259  wcel 1409  Vcvv 2574   cint 3643
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-v 2576  df-int 3644
This theorem is referenced by:  elint2  3650  elintab  3654  intss1  3658  intss  3664  intun  3674  intpr  3675  peano1  4345
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