Theorem ineq2 3399

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 3398	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
2		incom 3396	. 2 |- ( C i^i A ) = ( A i^i C )
3		incom 3396	. 2 |- ( C i^i B ) = ( B i^i C )
4	1, 2, 3	3eqtr4g 2287	1 |- ( A = B -> ( C i^i A ) = ( C i^i B ) )

Syntax hints:   -> wi 4   = wceq 1395   i^i cin 3196

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203

This theorem is referenced by:  ineq12  3400  ineq2i  3402  ineq2d  3405  uneqin  3455  intprg  3956  fiintim  7093  uzin2  11498  inopn  14677  basis1  14721  basis2  14722  baspartn  14724  metreslem  15054  qtopbasss  15195