Theorem ineq1i 3334

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

Hypothesis

Ref	Expression
ineq1i.1	|- A = B

Assertion

Ref	Expression
ineq1i	|- ( A i^i C ) = ( B i^i C )

Proof of Theorem ineq1i

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 |- A = B
2		ineq1 3331	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
3	1, 2	ax-mp 5	1 |- ( A i^i C ) = ( B i^i C )

Syntax hints:   = wceq 1353   i^i cin 3130

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137

This theorem is referenced by:  in12  3348  inindi  3354  dfrab2  3412  dfrab3  3413  disjpr2  3658  resres  4921  imainrect  5076  ssenen  6853  minmax  11240  xrminmax  11275  nnmindc  12037  nnminle  12038  setsfun  12499  setsfun0  12500  ressressg  12536  tgrest  13754