Theorem ineq2 3266

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 3265	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
2		incom 3263	. 2 |- ( C i^i A ) = ( A i^i C )
3		incom 3263	. 2 |- ( C i^i B ) = ( B i^i C )
4	1, 2, 3	3eqtr4g 2195	1 |- ( A = B -> ( C i^i A ) = ( C i^i B ) )

Syntax hints:   -> wi 4   = wceq 1331   i^i cin 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072

This theorem is referenced by:  ineq12  3267  ineq2i  3269  ineq2d  3272  uneqin  3322  intprg  3799  fiintim  6810  uzin2  10752  inopn  12159  basis1  12203  basis2  12204  baspartn  12206  metreslem  12538  qtopbasss  12679