Theorem eqeq12 2209

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 2203	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2		eqeq2 2206	. 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 1364

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 1461  ax-gen 1463  ax-4 1524  ax-17 1540  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-cleq 2189

This theorem is referenced by:  eqeq12i  2210  eqeq12d  2211  eqeqan12d  2212  funopg  5293  tfri3  6434  th3qlem1  6705  xpdom2  6899  difinfsnlem  7174  difinfsn  7175  xrlttri3  9889  bcn1  10867  summodc  11565  prodmodc  11760  ringinvnz1ne0  13681