Theorem preq2 3711

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 3710	. 2 |- ( A = B -> { A , C } = { B , C } )
2		prcom 3709	. 2 |- { C , A } = { A , C }
3		prcom 3709	. 2 |- { C , B } = { B , C }
4	1, 2, 3	3eqtr4g 2263	1 |- ( A = B -> { C , A } = { C , B } )

Syntax hints:   -> wi 4   = wceq 1373   {cpr 3634

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

This theorem is referenced by:  preq12  3712  preq2i  3714  preq2d  3717  tpeq2  3720  preq12bg  3814  opeq2  3820  uniprg  3865  intprg  3918  prexg  4255  opth  4281  opeqsn  4297  relop  4828  funopg  5305  en2  6912  prfidceq  7025  pr2ne  7300  hashprg  10953  bj-prexg  15847