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Theorem ineq12i 3245
Description: Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)
Hypotheses
Ref Expression
ineq1i.1  |-  A  =  B
ineq12i.2  |-  C  =  D
Assertion
Ref Expression
ineq12i  |-  ( A  i^i  C )  =  ( B  i^i  D
)

Proof of Theorem ineq12i
StepHypRef Expression
1 ineq1i.1 . 2  |-  A  =  B
2 ineq12i.2 . 2  |-  C  =  D
3 ineq12 3242 . 2  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  i^i  C
)  =  ( B  i^i  D ) )
41, 2, 3mp2an 422 1  |-  ( A  i^i  C )  =  ( B  i^i  D
)
Colors of variables: wff set class
Syntax hints:    = wceq 1316    i^i cin 3040
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047
This theorem is referenced by:  undir  3296  difindir  3301  inrab  3318  inrab2  3319  inxp  4643  resindi  4804  resindir  4805  cnvin  4916  rnin  4918  inimass  4925  funtp  5146  imainlem  5174  imain  5175  offres  6001  djuinr  6916  djuin  6917  casefun  6938  exmidfodomrlemim  7025  enq0enq  7207  explecnv  11242
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