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Theorem oteq3 4198
Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
oteq3  |-  ( A  =  B  ->  <. C ,  D ,  A >.  = 
<. C ,  D ,  B >. )

Proof of Theorem oteq3
StepHypRef Expression
1 opeq2 4188 . 2  |-  ( A  =  B  ->  <. <. C ,  D >. ,  A >.  = 
<. <. C ,  D >. ,  B >. )
2 df-ot 4007 . 2  |-  <. C ,  D ,  A >.  = 
<. <. C ,  D >. ,  A >.
3 df-ot 4007 . 2  |-  <. C ,  D ,  B >.  = 
<. <. C ,  D >. ,  B >.
41, 2, 33eqtr4g 2488 1  |-  ( A  =  B  ->  <. C ,  D ,  A >.  = 
<. C ,  D ,  B >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   <.cop 4004   <.cotp 4006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-ot 4007
This theorem is referenced by:  oteq3d  4201  otsndisj  4725  otiunsndisj  4726  efgi0  17369  efgi1  17370  mapdhcl  35264  mapdh6dN  35276  mapdh8  35326  mapdh9a  35327  mapdh9aOLDN  35328  hdmap1l6d  35351  hdmapval  35368  hdmapval2  35372  hdmapval3N  35378  otiunsndisjX  38871
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