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Theorem oteq1 4140
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 4131 . . 3  |-  ( A  =  B  ->  <. A ,  C >.  =  <. B ,  C >. )
21opeq1d 4137 . 2  |-  ( A  =  B  ->  <. <. A ,  C >. ,  D >.  = 
<. <. B ,  C >. ,  D >. )
3 df-ot 3953 . 2  |-  <. A ,  C ,  D >.  = 
<. <. A ,  C >. ,  D >.
4 df-ot 3953 . 2  |-  <. B ,  C ,  D >.  = 
<. <. B ,  C >. ,  D >.
52, 3, 43eqtr4g 2448 1  |-  ( A  =  B  ->  <. A ,  C ,  D >.  = 
<. B ,  C ,  D >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399   <.cop 3950   <.cotp 3952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-ot 3953
This theorem is referenced by:  oteq1d  4143  otiunsndisj  4667  efgi  16854  efgtf  16857  efgtval  16858  otiunsndisjX  32622  mapdh9a  37930  mapdh9aOLDN  37931  hdmapval2  37975
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