Theorem ineq1i 3428

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 3425	. 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 )

This proof depends on syntax axioms:   = 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:  in12  3442  inindi  3448  dfrab2  3508  dfrab3  3509  disjpr2  3773  resres  5075  imainrect  5233  ssenen  7152  minmax  12011  xrminmax  12047  nnmindc  12827  nnminle  12828  ballotfilem2  13277  ballotfilemfval0  13284  setsfun  13436  setsfun0  13437  ressressg  13478  tgrest  15319  ppi1  16176  ppiqub  16194