Theorem preq2 3672

Description: Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
preq2	|- ( A = B -> { C , A } = { C , B } )

Proof of Theorem preq2

Step	Hyp	Ref	Expression
1		preq1 3671	. 2 |- ( A = B -> { A , C } = { B , C } )
2		prcom 3670	. 2 |- { C , A } = { A , C }
3		prcom 3670	. 2 |- { C , B } = { B , C }
4	1, 2, 3	3eqtr4g 2235	1 |- ( A = B -> { C , A } = { C , B } )

Syntax hints:   -> wi 4   = wceq 1353   {cpr 3595

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601

This theorem is referenced by:  preq12  3673  preq2i  3675  preq2d  3678  tpeq2  3681  preq12bg  3775  opeq2  3781  uniprg  3826  intprg  3879  prexg  4213  opth  4239  opeqsn  4254  relop  4779  funopg  5252  pr2ne  7193  hashprg  10790  bj-prexg  14748