Theorem eqeq12 2052

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 2046	. 2 |- ( A = B -> ( A = C <-> B = C ) )
2		eqeq2 2049	. 2 |- ( C = D -> ( B = C <-> B = D ) )
3	1, 2	sylan9bb 435	1 |- ( ( A = B /\ C = D ) -> ( A = C <-> B = D ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-17 1419  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033

This theorem is referenced by:  eqeq12i  2053  eqeq12d  2054  eqeqan12d  2055  funopg  4934  tfri3  5953  th3qlem1  6208  xpdom2  6305  xrlttri3  8718