Theorem ineq2 3416

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 3415	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
2		incom 3411	. 2 |- ( C i^i A ) = ( A i^i C )
3		incom 3411	. 2 |- ( C i^i B ) = ( B i^i C )
4	1, 2, 3	3eqtr4g 2290	1 |- ( A = B -> ( C i^i A ) = ( C i^i B ) )

Syntax hints:   -> wi 4   = wceq 1398   i^i cin 3210

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217

This theorem is referenced by:  ineq12  3417  ineq2i  3419  ineq2d  3422  uneqin  3472  intprg  3982  fiintim  7191  uzin2  11672  inopn  14868  basis1  14912  basis2  14913  baspartn  14915  metreslem  15245  qtopbasss  15386