Theorem tpeq3 3784

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

Assertion

Ref	Expression
tpeq3	|- ( A = B -> { C , D , A } = { C , D , B } )

Proof of Theorem tpeq3

Step	Hyp	Ref	Expression
1		sneq 3705	. . 3 |- ( A = B -> { A } = { B } )
2	1	uneq2d 3377	. 2 |- ( A = B -> ( { C , D } u. { A } ) = ( { C , D } u. { B } ) )
3		df-tp 3702	. 2 |- { C , D , A } = ( { C , D } u. { A } )
4		df-tp 3702	. 2 |- { C , D , B } = ( { C , D } u. { B } )
5	2, 3, 4	3eqtr4g 2292	1 |- ( A = B -> { C , D , A } = { C , D , B } )

Syntax hints:   -> wi 4   = wceq 1398   u. cun 3212   {csn 3694   {cpr 3695   {ctp 3696

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-sn 3700  df-tp 3702

This theorem is referenced by:  tpeq3d  3787  tppreq3  3799  fztpval  10439