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

Theorem oteq3d 3990
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 3987 . 2  |-  ( A  =  B  ->  <. C ,  D ,  A >.  = 
<. C ,  D ,  B >. )
31, 2syl 16 1  |-  ( ph  -> 
<. C ,  D ,  A >.  =  <. C ,  D ,  B >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   <.cotp 3810
This theorem is referenced by:  oteq123d  3991  idafval  14204  coafval  14211  arwlid  14219  arwrid  14220  arwass  14221  efgi  15343  efgtf  15346  efgtval  15347  efgval2  15348  mapdh6bN  32472  mapdh6cN  32473  mapdh6dN  32474  mapdh6gN  32477  hdmap1l6b  32547  hdmap1l6c  32548  hdmap1l6d  32549  hdmap1l6g  32552
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-ot 3816
  Copyright terms: Public domain W3C validator