Theorem eqeq12 2178

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 2172	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2		eqeq2 2175	. 2 |- ( C = D -> ( B = C <-> B = D ) )
3	1, 2	sylan9bb 458	1 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343

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 1435  ax-gen 1437  ax-4 1498  ax-17 1514  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-cleq 2158

This theorem is referenced by:  eqeq12i  2179  eqeq12d  2180  eqeqan12d  2181  funopg  5222  tfri3  6335  th3qlem1  6603  xpdom2  6797  difinfsnlem  7064  difinfsn  7065  xrlttri3  9733  bcn1  10671  summodc  11324  prodmodc  11519