Theorem ineq12 3346

Description: Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.)

Assertion

Ref	Expression
ineq12	|- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )

Proof of Theorem ineq12

Step	Hyp	Ref	Expression
1		ineq1 3344	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
2		ineq2 3345	. 2 |- ( C = D -> ( B i^i C ) = ( B i^i D ) )
3	1, 2	sylan9eq 2242	1 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   i^i cin 3143

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150

This theorem is referenced by:  ineq12i  3349  ineq12d  3352  ineqan12d  3353  fnun  5337  endisj  6842  sbthlemi8  6981  pm54.43  7207  epttop  13974  restbasg  14052  txbas  14142  bj-inex  15043