Theorem preq2 3721

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 3720	. 2 |- ( A = B -> { A , C } = { B , C } )
2		prcom 3719	. 2 |- { C , A } = { A , C }
3		prcom 3719	. 2 |- { C , B } = { B , C }
4	1, 2, 3	3eqtr4g 2265	1 |- ( A = B -> { C , A } = { C , B } )

Syntax hints:   -> wi 4   = wceq 1373   {cpr 3644

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650

This theorem is referenced by:  preq12  3722  preq2i  3724  preq2d  3727  tpeq2  3730  preq12bg  3827  opeq2  3834  uniprg  3879  intprg  3932  prexg  4271  opth  4299  opeqsn  4315  relop  4846  funopg  5324  en2  6936  prfidceq  7051  pr2ne  7326  pr1or2  7328  hashprg  10990  upgrex  15814  bj-prexg  16046