Theorem eqeq12 2220

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 2214	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2		eqeq2 2217	. 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 1373

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 1471  ax-gen 1473  ax-4 1534  ax-17 1550  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-cleq 2200

This theorem is referenced by:  eqeq12i  2221  eqeq12d  2222  eqeqan12d  2223  funopg  5324  tfri3  6476  th3qlem1  6747  xpdom2  6951  difinfsnlem  7227  difinfsn  7228  xrlttri3  9954  bcn1  10940  summodc  11809  prodmodc  12004  ringinvnz1ne0  13926