Theorem ineq12 3369

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 3367	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
2		ineq2 3368	. 2 |- ( C = D -> ( B i^i C ) = ( B i^i D ) )
3	1, 2	sylan9eq 2258	1 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   i^i cin 3165

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172

This theorem is referenced by:  ineq12i  3372  ineq12d  3375  ineqan12d  3376  fnun  5382  endisj  6919  sbthlemi8  7066  pm54.43  7298  epttop  14562  restbasg  14640  txbas  14730  bj-inex  15847