Theorem ineq2i 3429

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

Hypothesis

Ref	Expression
ineq1i.1	|- A = B

Assertion

Ref	Expression
ineq2i	|- ( C i^i A ) = ( C i^i B )

Proof of Theorem ineq2i

Step	Hyp	Ref	Expression
1		ineq1i.1	. 2 |- A = B
2		ineq2 3426	. 2 |- ( A = B -> ( C i^i A ) = ( C i^i B ) )
3	1, 2	ax-mp 5	1 |- ( C i^i A ) = ( C i^i B )

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:  in4  3447  inindir  3449  indif2  3475  difun1  3491  dfrab3ss  3511  dfif3  3654  intunsn  4008  rint0  4009  riin0  4084  res0  5069  resres  5077  resundi  5078  resindi  5080  inres  5082  resiun2  5085  resopab  5110  dfse2  5163  dminxp  5235  imainrect  5236  resdmres  5282  funimacnv  5460  unfiin  7237  sbthlemi5  7283  dmaddpi  7698  dmmulpi  7699  hashtpgim  11319  fsumiun  12269  ressval2  13479  ressval3d  13485  lgsquadlem3  16403