Theorem tpeq2 3720

Description: Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Assertion

Ref	Expression
tpeq2	|- ( A = B -> { C , A , D } = { C , B , D } )

Proof of Theorem tpeq2

Step	Hyp	Ref	Expression
1		preq2 3711	. . 3 |- ( A = B -> { C , A } = { C , B } )
2	1	uneq1d 3326	. 2 |- ( A = B -> ( { C , A } u. { D } ) = ( { C , B } u. { D } ) )
3		df-tp 3641	. 2 |- { C , A , D } = ( { C , A } u. { D } )
4		df-tp 3641	. 2 |- { C , B , D } = ( { C , B } u. { D } )
5	2, 3, 4	3eqtr4g 2263	1 |- ( A = B -> { C , A , D } = { C , B , D } )

Syntax hints:   -> wi 4   = wceq 1373   u. cun 3164   {csn 3633   {cpr 3634   {ctp 3635

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-tp 3641

This theorem is referenced by:  tpeq2d  3723  fztpval  10205