Theorem tppreq3 3735

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 3720	. . 3 |- ( C = B -> { A , B , C } = { A , B , B } )
2	1	eqcoms 2207	. 2 |- ( B = C -> { A , B , C } = { A , B , B } )
3		tpidm23 3733	. 2 |- { A , B , B } = { A , B }
4	2, 3	eqtrdi 2253	1 |- ( B = C -> { A , B , C } = { A , B } )

Syntax hints:   -> wi 4   = wceq 1372   {cpr 3633   {ctp 3634

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3or 981  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-tp 3640

This theorem is referenced by: (None)