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Theorem uneq1d 3142
Description: Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
Hypothesis
Ref Expression
uneq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
uneq1d  |-  ( ph  ->  ( A  u.  C
)  =  ( B  u.  C ) )

Proof of Theorem uneq1d
StepHypRef Expression
1 uneq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 uneq1 3136 . 2  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
31, 2syl 14 1  |-  ( ph  ->  ( A  u.  C
)  =  ( B  u.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    u. cun 2986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992
This theorem is referenced by:  ifeq1  3382  preq1  3501  tpeq1  3510  tpeq2  3511  resasplitss  5146  fmptpr  5445  rdgisucinc  6097  oasuc  6172  omsuc  6180  fisseneq  6585  sbthlemi5  6606  exmidfodomrlemim  6763  fzpred  9406  fseq1p1m1  9430  nn0split  9467  nnsplit  9468  fzo0sn0fzo1  9552  fzosplitprm1  9565  zsupcllemstep  10807
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