Theorem tpeq2 3762

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

Assertion

Ref	Expression
tpeq2	|- ( A = B -> { C , A , D } = { C , B , D } )

Proof of Theorem tpeq2

Step	Hyp	Ref	Expression
1		preq2 3753	. . 3 |- ( A = B -> { C , A } = { C , B } )
2	1	uneq1d 3362	. 2 |- ( A = B -> ( { C , A } u. { D } ) = ( { C , B } u. { D } ) )
3		df-tp 3681	. 2 |- { C , A , D } = ( { C , A } u. { D } )
4		df-tp 3681	. 2 |- { C , B , D } = ( { C , B } u. { D } )
5	2, 3, 4	3eqtr4g 2289	1 |- ( A = B -> { C , A , D } = { C , B , D } )

Syntax hints:   -> wi 4   = wceq 1398   u. cun 3199   {csn 3673   {cpr 3674   {ctp 3675

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-tp 3681

This theorem is referenced by:  tpeq2d  3765  fztpval  10380