Theorem eqeq12 2183

Description: Equality relationship among 4 classes. (Contributed by NM, 3-Aug-1994.)

Assertion

Ref	Expression
eqeq12	|- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )

Proof of Theorem eqeq12

Step	Hyp	Ref	Expression
1		eqeq1 2177	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2		eqeq2 2180	. 2 |- ( C = D -> ( B = C <-> B = D ) )
3	1, 2	sylan9bb 459	1 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348

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-5 1440  ax-gen 1442  ax-4 1503  ax-17 1519  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163

This theorem is referenced by:  eqeq12i  2184  eqeq12d  2185  eqeqan12d  2186  funopg  5232  tfri3  6346  th3qlem1  6615  xpdom2  6809  difinfsnlem  7076  difinfsn  7077  xrlttri3  9754  bcn1  10692  summodc  11346  prodmodc  11541