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Theorem oteq123d 3827
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 3824 . 2  |-  ( ph  -> 
<. A ,  C ,  E >.  =  <. B ,  C ,  E >. )
3 oteq123d.2 . . 3  |-  ( ph  ->  C  =  D )
43oteq2d 3825 . 2  |-  ( ph  -> 
<. B ,  C ,  E >.  =  <. B ,  D ,  E >. )
5 oteq123d.3 . . 3  |-  ( ph  ->  E  =  F )
65oteq3d 3826 . 2  |-  ( ph  -> 
<. B ,  D ,  E >.  =  <. B ,  D ,  F >. )
72, 4, 63eqtrd 2332 1  |-  ( ph  -> 
<. A ,  C ,  E >.  =  <. B ,  D ,  F >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   <.cotp 3657
This theorem is referenced by:  idaval  13906  coaval  13916
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-ot 3663
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