Theorem preq2 3601

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 3600	. 2 |- ( A = B -> { A , C } = { B , C } )
2		prcom 3599	. 2 |- { C , A } = { A , C }
3		prcom 3599	. 2 |- { C , B } = { B , C }
4	1, 2, 3	3eqtr4g 2197	1 |- ( A = B -> { C , A } = { C , B } )

Syntax hints:   -> wi 4   = wceq 1331   {cpr 3528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534

This theorem is referenced by:  preq12  3602  preq2i  3604  preq2d  3607  tpeq2  3610  preq12bg  3700  opeq2  3706  uniprg  3751  intprg  3804  prexg  4133  opth  4159  opeqsn  4174  relop  4689  funopg  5157  pr2ne  7048  hashprg  10554  bj-prexg  13109