Theorem preq12 3602

Description: Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Assertion

Ref	Expression
preq12	|- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )

Proof of Theorem preq12

Step	Hyp	Ref	Expression
1		preq1 3600	. 2 |- ( A = C -> { A , B } = { C , B } )
2		preq2 3601	. 2 |- ( B = D -> { C , B } = { C , D } )
3	1, 2	sylan9eq 2192	1 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )

Syntax hints:   -> wi 4   /\ wa 103   = 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:  preq12i  3605  preq12d  3608  preq12b  3697  opthreg  4471  relop  4689  qtopbasss  12690