Theorem ineq12 3403

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 3401	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
2		ineq2 3402	. 2 |- ( C = D -> ( B i^i C ) = ( B i^i D ) )
3	1, 2	sylan9eq 2284	1 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   i^i cin 3199

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206

This theorem is referenced by:  ineq12i  3406  ineq12d  3409  ineqan12d  3410  fnun  5438  endisj  7007  sbthlemi8  7162  pm54.43  7394  epttop  14813  restbasg  14891  txbas  14981  bj-inex  16502