MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oteq3d Structured version   Unicode version

Theorem oteq3d 4233
Description: Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
oteq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
oteq3d  |-  ( ph  -> 
<. C ,  D ,  A >.  =  <. C ,  D ,  B >. )

Proof of Theorem oteq3d
StepHypRef Expression
1 oteq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 oteq3 4230 . 2  |-  ( A  =  B  ->  <. C ,  D ,  A >.  = 
<. C ,  D ,  B >. )
31, 2syl 16 1  |-  ( ph  -> 
<. C ,  D ,  A >.  =  <. C ,  D ,  B >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   <.cotp 4041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-ot 4042
This theorem is referenced by:  oteq123d  4234  idafval  15259  coafval  15266  arwlid  15274  arwrid  15275  arwass  15276  efgi  16610  efgtf  16613  efgtval  16614  efgval2  16615  mapdh6bN  36935  mapdh6cN  36936  mapdh6dN  36937  mapdh6gN  36940  hdmap1l6b  37010  hdmap1l6c  37011  hdmap1l6d  37012  hdmap1l6g  37015
  Copyright terms: Public domain W3C validator