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Theorem oteq123d 3999
Description: Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
oteq1d.1  |-  ( ph  ->  A  =  B )
oteq123d.2  |-  ( ph  ->  C  =  D )
oteq123d.3  |-  ( ph  ->  E  =  F )
Assertion
Ref Expression
oteq123d  |-  ( ph  -> 
<. A ,  C ,  E >.  =  <. B ,  D ,  F >. )

Proof of Theorem oteq123d
StepHypRef Expression
1 oteq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21oteq1d 3996 . 2  |-  ( ph  -> 
<. A ,  C ,  E >.  =  <. B ,  C ,  E >. )
3 oteq123d.2 . . 3  |-  ( ph  ->  C  =  D )
43oteq2d 3997 . 2  |-  ( ph  -> 
<. B ,  C ,  E >.  =  <. B ,  D ,  E >. )
5 oteq123d.3 . . 3  |-  ( ph  ->  E  =  F )
65oteq3d 3998 . 2  |-  ( ph  -> 
<. B ,  D ,  E >.  =  <. B ,  D ,  F >. )
72, 4, 63eqtrd 2472 1  |-  ( ph  -> 
<. A ,  C ,  E >.  =  <. B ,  D ,  F >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   <.cotp 3818
This theorem is referenced by:  idaval  14213  coaval  14223  matval  27442  el2wlkonotot0  28339
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 2417
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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-ot 3824
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