Theorem ineq2 3426

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 3425	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
2		incom 3421	. 2 |- ( C i^i A ) = ( A i^i C )
3		incom 3421	. 2 |- ( C i^i B ) = ( B i^i C )
4	1, 2, 3	3eqtr4g 2296	1 |- ( A = B -> ( C i^i A ) = ( C i^i B ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   i^i cin 3219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226

This theorem is used by:  ineq12  3427  ineq2i  3429  ineq2d  3432  uneqin  3482  intprg  4003  fiintim  7238  uzin2  11753  ballotfilemfval  13229  ballotfilemgval  13267  inopn  15104  basis1  15148  basis2  15149  baspartn  15151  metreslem  15481  qtopbasss  15622