Theorem eqeq12 2101

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 2095	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2		eqeq2 2098	. 2 |- ( C = D -> ( B = C <-> B = D ) )
3	1, 2	sylan9bb 451	1 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1290

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-4 1446  ax-17 1465  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-cleq 2082

This theorem is referenced by:  eqeq12i  2102  eqeq12d  2103  eqeqan12d  2104  funopg  5061  tfri3  6146  th3qlem1  6408  xpdom2  6601  xrlttri3  9328  bcn1  10227  isummo  10834