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Theorem ineq12i 3164
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 3161 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
41, 2, 3mp2an 410 1 (𝐴𝐶) = (𝐵𝐷)
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cin 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952
This theorem is referenced by:  undir  3215  difindir  3220  inrab  3237  inrab2  3238  inxp  4498  resindi  4655  resindir  4656  cnvin  4759  rnin  4761  inimass  4768  funtp  4980  imainlem  5008  imain  5009  offres  5790  enq0enq  6587
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