MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coeq12i Structured version   Visualization version   GIF version

Theorem coeq12i 5727
Description: Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
Hypotheses
Ref Expression
coeq12i.1 𝐴 = 𝐵
coeq12i.2 𝐶 = 𝐷
Assertion
Ref Expression
coeq12i (𝐴𝐶) = (𝐵𝐷)

Proof of Theorem coeq12i
StepHypRef Expression
1 coeq12i.1 . . 3 𝐴 = 𝐵
21coeq1i 5723 . 2 (𝐴𝐶) = (𝐵𝐶)
3 coeq12i.2 . . 3 𝐶 = 𝐷
43coeq2i 5724 . 2 (𝐵𝐶) = (𝐵𝐷)
52, 4eqtri 2841 1 (𝐴𝐶) = (𝐵𝐷)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  ccom 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-in 3940  df-ss 3949  df-br 5058  df-opab 5120  df-co 5557
This theorem is referenced by:  madetsumid  20998  mdetleib2  21125  imsval  28389  pjcmul1i  29905  coprprop  30361  cotrcltrcl  39948  brtrclfv2  39950  clsneif1o  40332
  Copyright terms: Public domain W3C validator