Theorem tpeq1 3755

Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Assertion

Ref	Expression
tpeq1	|- ( A = B -> { A , C , D } = { B , C , D } )

Proof of Theorem tpeq1

Step	Hyp	Ref	Expression
1		preq1 3746	. . 3 |- ( A = B -> { A , C } = { B , C } )
2	1	uneq1d 3358	. 2 |- ( A = B -> ( { A , C } u. { D } ) = ( { B , C } u. { D } ) )
3		df-tp 3675	. 2 |- { A , C , D } = ( { A , C } u. { D } )
4		df-tp 3675	. 2 |- { B , C , D } = ( { B , C } u. { D } )
5	2, 3, 4	3eqtr4g 2287	1 |- ( A = B -> { A , C , D } = { B , C , D } )

Syntax hints:   -> wi 4   = wceq 1395   u. cun 3196   {csn 3667   {cpr 3668   {ctp 3669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-pr 3674  df-tp 3675

This theorem is referenced by:  tpeq1d  3758