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Theorem ineq2 3160
Description: Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)
Assertion
Ref Expression
ineq2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem ineq2
StepHypRef Expression
1 ineq1 3159 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 incom 3157 . 2 (𝐶𝐴) = (𝐴𝐶)
3 incom 3157 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33eqtr4g 2113 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = 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:  ineq12  3161  ineq2i  3163  ineq2d  3166  uneqin  3216  intprg  3676  uzin2  9814
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