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Theorem uneq1d 3090
 Description: Deduction adding union to the right in a class equality. (Contributed by NM, 29-Mar-1998.)
Hypothesis
Ref Expression
uneq1d.1 (φA = B)
Assertion
Ref Expression
uneq1d (φ → (A𝐶) = (B𝐶))

Proof of Theorem uneq1d
StepHypRef Expression
1 uneq1d.1 . 2 (φA = B)
2 uneq1 3084 . 2 (A = B → (A𝐶) = (B𝐶))
31, 2syl 14 1 (φ → (A𝐶) = (B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∪ cun 2909 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916 This theorem is referenced by:  ifeq1  3328  preq1  3438  tpeq1  3447  tpeq2  3448  resasplitss  5012  fmptpr  5298  rdgisucinc  5912  oasuc  5983  omsuc  5990  fzpred  8682  fseq1p1m1  8706  nn0split  8744  fzo0sn0fzo1  8827  fzosplitprm1  8840
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