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Theorem elint 3772
 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 2200 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32imbi2d 229 . . 3 (𝑦 = 𝐴 → ((𝑥𝐵𝑦𝑥) ↔ (𝑥𝐵𝐴𝑥)))
43albidv 1796 . 2 (𝑦 = 𝐴 → (∀𝑥(𝑥𝐵𝑦𝑥) ↔ ∀𝑥(𝑥𝐵𝐴𝑥)))
5 df-int 3767 . 2 𝐵 = {𝑦 ∣ ∀𝑥(𝑥𝐵𝑦𝑥)}
61, 4, 5elab2 2827 1 (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329   = wceq 1331   ∈ wcel 1480  Vcvv 2681  ∩ cint 3766 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 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-int 3767 This theorem is referenced by:  elint2  3773  elintab  3777  intss1  3781  intss  3787  intun  3797  intpr  3798  peano1  4503
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