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Theorem ineq1d 3276
Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)
Hypothesis
Ref Expression
ineq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
ineq1d (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem ineq1d
StepHypRef Expression
1 ineq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 ineq1 3270 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
31, 2syl 14 1 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
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:  diftpsn3  3661  disji2  3922  ordpwsucexmid  4485  riinint  4800  fnresdisj  5233  fnimadisj  5243  ecinxp  6504  fiintim  6817  fival  6858  fzval2  9800  fvinim0ffz  10025  fsum1p  11194  restopnb  12360  metrest  12685  qtopbasss  12700
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