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Theorem coeq2d 4669
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 4665 . 2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2syl 14 1 (𝜑 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  ccom 4511
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-in 3045  df-ss 3052  df-br 3898  df-opab 3958  df-co 4516
This theorem is referenced by:  coeq12d  4671  relcoi1  5038  f1ococnv1  5362  funcoeqres  5364  fcof1o  5656  foeqcnvco  5657  mapen  6706  hashfacen  10519
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