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Theorem uneq2d 3127
Description: Deduction adding union to the left in a class equality. (Contributed by NM, 29-Mar-1998.)
Hypothesis
Ref Expression
uneq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
uneq2d  |-  ( ph  ->  ( C  u.  A
)  =  ( C  u.  B ) )

Proof of Theorem uneq2d
StepHypRef Expression
1 uneq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 uneq2 3121 . 2  |-  ( A  =  B  ->  ( C  u.  A )  =  ( C  u.  B ) )
31, 2syl 14 1  |-  ( ph  ->  ( C  u.  A
)  =  ( C  u.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    u. cun 2972
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978
This theorem is referenced by:  ifeq2  3363  tpeq3  3488  iununir  3767  unisucg  4177  relcoi1  4879  resasplitss  5100  fvun1  5271  fmptapd  5386  fvunsng  5389  tfr1onlemaccex  5997  tfrcllemaccex  6010  rdgeq1  6020  rdgivallem  6030  rdgisuc1  6033  rdgon  6035  rdg0  6036  oav2  6107  oasuc  6108  omv2  6109  omsuc  6116  unsnfidcex  6440  undiffi  6443  fnfi  6446  pm54.43  6518  fzsuc  9162  fseq1p1m1  9187  fseq1m1p1  9188  fzosplitsnm1  9295  fzosplitsn  9319  fzosplitprm1  9320
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