Theorem tppreq3 3473

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 3458	. . 3 |- ( C = B -> { A , B , C } = { A , B , B } )
2	1	eqcoms 2043	. 2 |- ( B = C -> { A , B , C } = { A , B , B } )
3		tpidm23 3471	. 2 |- { A , B , B } = { A , B }
4	2, 3	syl6eq 2088	1 |- ( B = C -> { A , B , C } = { A , B } )

Syntax hints:   -> wi 4   = wceq 1243   {cpr 3376   {ctp 3377

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3or 886  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-tp 3383

This theorem is referenced by: (None)