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Theorem oteq1 4065
Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
oteq1  |-  ( A  =  B  ->  <. A ,  C ,  D >.  = 
<. B ,  C ,  D >. )

Proof of Theorem oteq1
StepHypRef Expression
1 opeq1 4056 . . 3  |-  ( A  =  B  ->  <. A ,  C >.  =  <. B ,  C >. )
21opeq1d 4062 . 2  |-  ( A  =  B  ->  <. <. A ,  C >. ,  D >.  = 
<. <. B ,  C >. ,  D >. )
3 df-ot 3883 . 2  |-  <. A ,  C ,  D >.  = 
<. <. A ,  C >. ,  D >.
4 df-ot 3883 . 2  |-  <. B ,  C ,  D >.  = 
<. <. B ,  C >. ,  D >.
52, 3, 43eqtr4g 2498 1  |-  ( A  =  B  ->  <. A ,  C ,  D >.  = 
<. B ,  C ,  D >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364   <.cop 3880   <.cotp 3882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-rab 2722  df-v 2972  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-ot 3883
This theorem is referenced by:  oteq1d  4068  efgi  16209  efgtf  16212  efgtval  16213  otiunsndisj  30041  otiunsndisjX  30042  mapdh9a  35123  mapdh9aOLDN  35124  hdmapval2  35168
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