MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpeq3d Structured version   Unicode version

Theorem tpeq3d 4093
Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
Hypothesis
Ref Expression
tpeq1d.1  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
tpeq3d  |-  ( ph  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )

Proof of Theorem tpeq3d
StepHypRef Expression
1 tpeq1d.1 . 2  |-  ( ph  ->  A  =  B )
2 tpeq3 4090 . 2  |-  ( A  =  B  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
31, 2syl 17 1  |-  ( ph  ->  { C ,  D ,  A }  =  { C ,  D ,  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   {ctp 4002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-un 3441  df-sn 3999  df-tp 4003
This theorem is referenced by:  tpeq123d  4094  erngset  34336  erngset-rN  34344
  Copyright terms: Public domain W3C validator