Theorem ineq1i 3378

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 3375	. 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 1373   i^i cin 3173

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-in 3180

This theorem is referenced by:  in12  3392  inindi  3398  dfrab2  3456  dfrab3  3457  disjpr2  3707  resres  4990  imainrect  5147  ssenen  6973  minmax  11656  xrminmax  11691  nnmindc  12470  nnminle  12471  setsfun  12982  setsfun0  12983  ressressg  13022  tgrest  14756