Theorem ineq1d 3431

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 3425	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   = 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:  diftpsn3  3856  disji2  4122  ordpwsucexmid  4717  riinint  5043  fnresdisj  5493  fnimadisj  5504  ecinxp  6884  fiintim  7238  fival  7304  fzval2  10424  fvinim0ffz  10670  hashfibc  11297  fsum1p  12201  fprod1p  12382  ballotfilemfval  13278  ballotfilemfp1  13280  ballotfilemfc0  13281  ballotfilemfcc  13282  ballotfilemgval  13316  ballotfilemgun  13317  strressid  13474  restopnb  15331  metrest  15656  qtopbasss  15671  ppiqval  16160  ppival2  16161  ppival2g  16162  ppiprm  16170  ppidif  16175