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Theorem coeq2i 5816
Description: Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
Hypothesis
Ref Expression
coeq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
coeq2i (𝐶𝐴) = (𝐶𝐵)

Proof of Theorem coeq2i
StepHypRef Expression
1 coeq1i.1 . 2 𝐴 = 𝐵
2 coeq2 5814 . 2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
31, 2ax-mp 5 1 (𝐶𝐴) = (𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  ccom 5637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3447  df-in 3917  df-ss 3927  df-br 5106  df-opab 5168  df-co 5642
This theorem is referenced by:  coeq12i  5819  cocnvcnv2  6210  co01  6213  dfpo2  6248  fcoi1  6716  f1ofvswap  7251  dftpos2  8173  tposco  8187  cottrcl  9654  canthp1  10589  cats1co  14744  isoval  17647  mvdco  19225  evlsval  21494  evl1fval1lem  21694  evl1var  21700  pf1ind  21719  imasdsf1olem  23724  hoico1  30645  hoid1i  30678  pjclem1  31084  pjclem3  31086  pjci  31089  cycpmconjv  31935  cycpmconjs  31949  poimirlem9  36077  cdlemk45  39400  cononrel1  41847  trclubgNEW  41871  trclrelexplem  41964  relexpaddss  41971  cotrcltrcl  41978  cortrcltrcl  41993  corclrtrcl  41994  cotrclrcl  41995  cortrclrcl  41996  cotrclrtrcl  41997  cortrclrtrcl  41998  brco3f1o  42286  clsneibex  42355  neicvgbex  42365  subsaliuncl  44570  meadjiun  44678  fundcmpsurinjimaid  45574
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