Theorem ineq2 3195

Description: Equality theorem for intersection of two classes. (Contributed by NM, 26-Dec-1993.)

Assertion

Ref	Expression
ineq2	|- ( A = B -> ( C i^i A ) = ( C i^i B ) )

Proof of Theorem ineq2

Step	Hyp	Ref	Expression
1		ineq1 3194	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
2		incom 3192	. 2 |- ( C i^i A ) = ( A i^i C )
3		incom 3192	. 2 |- ( C i^i B ) = ( B i^i C )
4	1, 2, 3	3eqtr4g 2145	1 |- ( A = B -> ( C i^i A ) = ( C i^i B ) )

Syntax hints:   -> wi 4   = wceq 1289   i^i cin 2998

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005

This theorem is referenced by:  ineq12  3196  ineq2i  3198  ineq2d  3201  uneqin  3250  intprg  3721  fiintim  6639  uzin2  10420  inopn  11600