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Theorem difeq1d 3163
Description: Deduction adding difference to the right in a class equality. (Contributed by NM, 15-Nov-2002.)
Hypothesis
Ref Expression
difeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
difeq1d (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem difeq1d
StepHypRef Expression
1 difeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 difeq1 3157 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
31, 2syl 14 1 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  cdif 3038
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-dif 3043
This theorem is referenced by:  difeq12d  3165  diftpsn3  3631  phplem4  6717  phplem3g  6718  phplem4on  6729  en2other2  7020  isstruct2im  11896  isstruct2r  11897  setsfun0  11922  cldval  12195  difopn  12204  cnclima  12319
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