Theorem ineq1i 3347

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 3344	. 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 1364   i^i cin 3143

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150

This theorem is referenced by:  in12  3361  inindi  3367  dfrab2  3425  dfrab3  3426  disjpr2  3671  resres  4934  imainrect  5089  ssenen  6869  minmax  11256  xrminmax  11291  nnmindc  12053  nnminle  12054  setsfun  12515  setsfun0  12516  ressressg  12553  tgrest  14053