Theorem tpeq1 3757

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 3748	. . 3 |- ( A = B -> { A , C } = { B , C } )
2	1	uneq1d 3360	. 2 |- ( A = B -> ( { A , C } u. { D } ) = ( { B , C } u. { D } ) )
3		df-tp 3677	. 2 |- { A , C , D } = ( { A , C } u. { D } )
4		df-tp 3677	. 2 |- { B , C , D } = ( { B , C } u. { D } )
5	2, 3, 4	3eqtr4g 2289	1 |- ( A = B -> { A , C , D } = { B , C , D } )

Syntax hints:   -> wi 4   = wceq 1397   u. cun 3198   {csn 3669   {cpr 3670   {ctp 3671

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-tp 3677

This theorem is referenced by:  tpeq1d  3760