Theorem tppreq3 3778

Description: An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)

Assertion

Ref	Expression
tppreq3	|- ( B = C -> { A , B , C } = { A , B } )

Proof of Theorem tppreq3

Step	Hyp	Ref	Expression
1		tpeq3 3763	. . 3 |- ( C = B -> { A , B , C } = { A , B , B } )
2	1	eqcoms 2234	. 2 |- ( B = C -> { A , B , C } = { A , B , B } )
3		tpidm23 3776	. 2 |- { A , B , B } = { A , B }
4	2, 3	eqtrdi 2280	1 |- ( B = C -> { A , B , C } = { A , B } )

Syntax hints:   -> wi 4   = wceq 1398   {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-3or 1006  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: (None)