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Theorem ineq12i 3200
 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 (𝐴𝐶) = (𝐵𝐷)

Proof of Theorem ineq12i
StepHypRef Expression
1 ineq1i.1 . 2 𝐴 = 𝐵
2 ineq12i.2 . 2 𝐶 = 𝐷
3 ineq12 3197 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
41, 2, 3mp2an 418 1 (𝐴𝐶) = (𝐵𝐷)
 Colors of variables: wff set class Syntax hints:   = wceq 1290   ∩ cin 2999 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071 This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-in 3006 This theorem is referenced by:  undir  3250  difindir  3255  inrab  3272  inrab2  3273  inxp  4583  resindi  4741  resindir  4742  cnvin  4852  rnin  4854  inimass  4861  funtp  5080  imainlem  5108  imain  5109  offres  5920  djuinr  6809  casefun  6830  exmidfodomrlemim  6881  enq0enq  7044  explecnv  10953
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