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Theorem coeq1d 4709
 Description: Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
Hypothesis
Ref Expression
coeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
coeq1d (𝜑 → (𝐴𝐶) = (𝐵𝐶))

Proof of Theorem coeq1d
StepHypRef Expression
1 coeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 coeq1 4705 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
31, 2syl 14 1 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∘ ccom 4552 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 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  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-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-in 3083  df-ss 3090  df-br 3939  df-opab 3999  df-co 4557 This theorem is referenced by:  coeq12d  4712  fcof1o  5699  mapen  6749  hashfacen  10631
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