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Theorem ineq2 3271
 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 3270 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 incom 3268 . 2 (𝐶𝐴) = (𝐴𝐶)
3 incom 3268 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33eqtr4g 2197 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∩ cin 3070 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077 This theorem is referenced by:  ineq12  3272  ineq2i  3274  ineq2d  3277  uneqin  3327  intprg  3804  fiintim  6817  uzin2  10766  inopn  12179  basis1  12223  basis2  12224  baspartn  12226  metreslem  12558  qtopbasss  12699
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