Theorem ineq1d 3373

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 3367	. 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 3165

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172

This theorem is referenced by:  diftpsn3  3774  disji2  4037  ordpwsucexmid  4618  riinint  4939  fnresdisj  5386  fnimadisj  5396  ecinxp  6697  fiintim  7028  fival  7072  fzval2  10133  fvinim0ffz  10370  fsum1p  11729  fprod1p  11910  strressid  12903  restopnb  14653  metrest  14978  qtopbasss  14993