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Theorem elinti 3615
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 3614 . . 3 (A B → (A Bx B A x))
2 eleq2 2098 . . . 4 (x = 𝐶 → (A xA 𝐶))
32rspccv 2647 . . 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 1390  wral 2300   cint 3606
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-int 3607
This theorem is referenced by: (None)
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