Theorem tppreq3 3774

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 3759	. . 3 |- ( C = B -> { A , B , C } = { A , B , B } )
2	1	eqcoms 2234	. 2 |- ( B = C -> { A , B , C } = { A , B , B } )
3		tpidm23 3772	. 2 |- { A , B , B } = { A , B }
4	2, 3	eqtrdi 2280	1 |- ( B = C -> { A , B , C } = { A , B } )

Syntax hints:   -> wi 4   = wceq 1397   {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-3or 1005  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: (None)