Theorem eqeq12 2218

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 2212	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2		eqeq2 2215	. 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 1470  ax-gen 1472  ax-4 1533  ax-17 1549  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198

This theorem is referenced by:  eqeq12i  2219  eqeq12d  2220  eqeqan12d  2221  funopg  5305  tfri3  6453  th3qlem1  6724  xpdom2  6926  difinfsnlem  7201  difinfsn  7202  xrlttri3  9919  bcn1  10903  summodc  11694  prodmodc  11889  ringinvnz1ne0  13811