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Theorem difeq2d 3199
Description: Deduction adding difference to the left in a class equality. (Contributed by NM, 15-Nov-2002.)
Hypothesis
Ref Expression
difeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
difeq2d (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem difeq2d
StepHypRef Expression
1 difeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 difeq2 3193 . 2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2syl 14 1 (𝜑 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1332  cdif 3073
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-in1 604  ax-in2 605  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-rab 2426  df-dif 3078
This theorem is referenced by:  difeq12d  3200  phplem3  6756  phplem4  6757  phplem3g  6758  phplem4dom  6764  phplem4on  6769  fidifsnen  6772  xpfi  6826  sbthlem2  6854  sbthlemi3  6855  isbth  6863  ismkvnex  7037  setsvalg  12028  setsvala  12029  exmid1stab  13368
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