Theorem tpeq3 3710

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 3633	. . 3 |- ( A = B -> { A } = { B } )
2	1	uneq2d 3317	. 2 |- ( A = B -> ( { C , D } u. { A } ) = ( { C , D } u. { B } ) )
3		df-tp 3630	. 2 |- { C , D , A } = ( { C , D } u. { A } )
4		df-tp 3630	. 2 |- { C , D , B } = ( { C , D } u. { B } )
5	2, 3, 4	3eqtr4g 2254	1 |- ( A = B -> { C , D , A } = { C , D , B } )

Syntax hints:   -> wi 4   = wceq 1364   u. cun 3155   {csn 3622   {cpr 3623   {ctp 3624

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-tp 3630

This theorem is referenced by:  tpeq3d  3713  tppreq3  3725  fztpval  10158