Theorem tpeq3 3558

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 3485	. . 3 |- ( A = B -> { A } = { B } )
2	1	uneq2d 3177	. 2 |- ( A = B -> ( { C , D } u. { A } ) = ( { C , D } u. { B } ) )
3		df-tp 3482	. 2 |- { C , D , A } = ( { C , D } u. { A } )
4		df-tp 3482	. 2 |- { C , D , B } = ( { C , D } u. { B } )
5	2, 3, 4	3eqtr4g 2157	1 |- ( A = B -> { C , D , A } = { C , D , B } )

Syntax hints:   -> wi 4   = wceq 1299   u. cun 3019   {csn 3474   {cpr 3475   {ctp 3476

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-tp 3482

This theorem is referenced by:  tpeq3d  3561  tppreq3  3573  fztpval  9704