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Theorem oteq123d 3811
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 3808 . 2  |-  ( ph  -> 
<. A ,  C ,  E >.  =  <. B ,  C ,  E >. )
3 oteq123d.2 . . 3  |-  ( ph  ->  C  =  D )
43oteq2d 3809 . 2  |-  ( ph  -> 
<. B ,  C ,  E >.  =  <. B ,  D ,  E >. )
5 oteq123d.3 . . 3  |-  ( ph  ->  E  =  F )
65oteq3d 3810 . 2  |-  ( ph  -> 
<. B ,  D ,  E >.  =  <. B ,  D ,  F >. )
72, 4, 63eqtrd 2319 1  |-  ( ph  -> 
<. A ,  C ,  E >.  =  <. B ,  D ,  F >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   <.cotp 3644
This theorem is referenced by:  idaval  13890  coaval  13900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650
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