Theorem eqeq12 2201

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 2195	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2		eqeq2 2198	. 2 |- ( C = D -> ( B = C <-> B = D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1363

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-5 1457  ax-gen 1459  ax-4 1520  ax-17 1536  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-cleq 2181

This theorem is referenced by:  eqeq12i  2202  eqeq12d  2203  eqeqan12d  2204  funopg  5264  tfri3  6385  th3qlem1  6654  xpdom2  6848  difinfsnlem  7115  difinfsn  7116  xrlttri3  9814  bcn1  10755  summodc  11408  prodmodc  11603  ringinvnz1ne0  13361