Theorem preq12 3673

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 3671	. 2 |- ( A = C -> { A , B } = { C , B } )
2		preq2 3672	. 2 |- ( B = D -> { C , B } = { C , D } )
3	1, 2	sylan9eq 2230	1 |- ( ( A = C /\ B = D ) -> { A , B } = { C , D } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   {cpr 3595

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601

This theorem is referenced by:  preq12i  3676  preq12d  3679  preq12b  3772  opthreg  4557  relop  4779  qtopbasss  14060