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Theorem tpeq123d 3673
Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)
Hypotheses
Ref Expression
tpeq1d.1  |-  ( ph  ->  A  =  B )
tpeq123d.2  |-  ( ph  ->  C  =  D )
tpeq123d.3  |-  ( ph  ->  E  =  F )
Assertion
Ref Expression
tpeq123d  |-  ( ph  ->  { A ,  C ,  E }  =  { B ,  D ,  F } )

Proof of Theorem tpeq123d
StepHypRef Expression
1 tpeq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21tpeq1d 3670 . 2  |-  ( ph  ->  { A ,  C ,  E }  =  { B ,  C ,  E } )
3 tpeq123d.2 . . 3  |-  ( ph  ->  C  =  D )
43tpeq2d 3671 . 2  |-  ( ph  ->  { B ,  C ,  E }  =  { B ,  D ,  E } )
5 tpeq123d.3 . . 3  |-  ( ph  ->  E  =  F )
65tpeq3d 3672 . 2  |-  ( ph  ->  { B ,  D ,  E }  =  { B ,  D ,  F } )
72, 4, 63eqtrd 2207 1  |-  ( ph  ->  { A ,  C ,  E }  =  { B ,  D ,  F } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348   {ctp 3583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3587  df-pr 3588  df-tp 3589
This theorem is referenced by:  fz0tp  10071  fz0to4untppr  10073  fzo0to3tp  10168
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