Theorem ineq1d 3381

Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.)

Hypothesis

Ref	Expression
ineq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
ineq1d	|- ( ph -> ( A i^i C ) = ( B i^i C ) )

Proof of Theorem ineq1d

Step	Hyp	Ref	Expression
1		ineq1d.1	. 2 |- ( ph -> A = B )
2		ineq1 3375	. 2 |- ( A = B -> ( A i^i C ) = ( B i^i C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A i^i C ) = ( B i^i C ) )

Syntax hints:   -> wi 4   = 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:  diftpsn3  3785  disji2  4051  ordpwsucexmid  4636  riinint  4958  fnresdisj  5405  fnimadisj  5416  ecinxp  6720  fiintim  7054  fival  7098  fzval2  10168  fvinim0ffz  10407  fsum1p  11844  fprod1p  12025  strressid  13018  restopnb  14768  metrest  15093  qtopbasss  15108