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Theorem oteq1d 4214
Description: Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypothesis
Ref Expression
oteq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
oteq1d  |-  ( ph  -> 
<. A ,  C ,  D >.  =  <. B ,  C ,  D >. )

Proof of Theorem oteq1d
StepHypRef Expression
1 oteq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 oteq1 4211 . 2  |-  ( A  =  B  ->  <. A ,  C ,  D >.  = 
<. B ,  C ,  D >. )
31, 2syl 16 1  |-  ( ph  -> 
<. A ,  C ,  D >.  =  <. B ,  C ,  D >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1383   <.cotp 4022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-ot 4023
This theorem is referenced by:  oteq123d  4217  msrfval  28875  msrid  28883  elmsta  28886  mthmpps  28920  hdmapfval  37432
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