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Theorem elinti 3598
 Description: Membership in class intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
elinti (A B → (𝐶 BA 𝐶))

Proof of Theorem elinti
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elintg 3597 . . 3 (A B → (A Bx B A x))
2 eleq2 2083 . . . 4 (x = 𝐶 → (A xA 𝐶))
32rspccv 2630 . . 3 (x B A x → (𝐶 BA 𝐶))
41, 3syl6bi 152 . 2 (A B → (A B → (𝐶 BA 𝐶)))
54pm2.43i 43 1 (A B → (𝐶 BA 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1374  ∀wral 2284  ∩ cint 3589 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-int 3590 This theorem is referenced by: (None)
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