Theorem ineq12 3178

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 3176	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
2		ineq2 3177	. 2 |- ( C = D -> ( B i^i C ) = ( B i^i D ) )
3	1, 2	sylan9eq 2135	1 |- ( ( A = B /\ C = D ) -> ( A i^i C ) = ( B i^i D ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   i^i cin 2981

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-in 2988

This theorem is referenced by:  ineq12i  3181  ineq12d  3184  ineqan12d  3185  fnun  5056  endisj  6390  pm54.43  6588  bj-inex  10983