Theorem preq2 3682

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 3681	. 2 |- ( A = B -> { A , C } = { B , C } )
2		prcom 3680	. 2 |- { C , A } = { A , C }
3		prcom 3680	. 2 |- { C , B } = { B , C }
4	1, 2, 3	3eqtr4g 2245	1 |- ( A = B -> { C , A } = { C , B } )

Syntax hints:   -> wi 4   = wceq 1363   {cpr 3605

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611

This theorem is referenced by:  preq12  3683  preq2i  3685  preq2d  3688  tpeq2  3691  preq12bg  3785  opeq2  3791  uniprg  3836  intprg  3889  prexg  4223  opth  4249  opeqsn  4264  relop  4789  funopg  5262  pr2ne  7204  hashprg  10801  bj-prexg  14934