Theorem tppreq3 3621

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 3606	. . 3 |- ( C = B -> { A , B , C } = { A , B , B } )
2	1	eqcoms 2140	. 2 |- ( B = C -> { A , B , C } = { A , B , B } )
3		tpidm23 3619	. 2 |- { A , B , B } = { A , B }
4	2, 3	syl6eq 2186	1 |- ( B = C -> { A , B , C } = { A , B } )

Syntax hints:   -> wi 4   = wceq 1331   {cpr 3523   {ctp 3524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3or 963  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-tp 3530

This theorem is referenced by: (None)