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Theorem coeq2d 4741
 Description: Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
Hypothesis
Ref Expression
coeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
coeq2d (𝜑 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem coeq2d
StepHypRef Expression
1 coeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 coeq2 4737 . 2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2syl 14 1 (𝜑 → (𝐶𝐴) = (𝐶𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∘ ccom 4583 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-in 3104  df-ss 3111  df-br 3962  df-opab 4022  df-co 4588 This theorem is referenced by:  coeq12d  4743  relcoi1  5110  f1ococnv1  5436  funcoeqres  5438  fcof1o  5730  foeqcnvco  5731  mapen  6780  hashfacen  10684
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