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Theorem coeq1d 4545
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 4541 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
31, 2syl 14 1 (𝜑 → (𝐴𝐶) = (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  ccom 4395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-in 2988  df-ss 2995  df-br 3806  df-opab 3860  df-co 4400
This theorem is referenced by:  coeq12d  4548  fcof1o  5480
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