Theorem ineq1d 2216

Description: Equality deduction for intersection of two classes.

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 2210	. 2 |- (A = B -> (A i^i C) = (B i^i C))
3	1, 2	syl 10	1 |- (ph -> (A i^i C) = (B i^i C))

Syntax hints:   -> wi 3   = wceq 956   i^i cin 2046

This theorem is referenced by:  ineq12d 2218  fnresdisj 3597  funimadisj 3606  fiint 4559  fiintOLD 4560  kmlem12 4776  limsupvalt 6529  subtop 7646  indistop 7648  bcthlem15 8013  chdmj2t 9453  cmcmlem 9534  pjoml2t 9554  fh2t 9562  mdbrt 10221  mdit 10222  mdbr3 10224  mdbr4 10225  dmdmdt 10227  dmdbr3 10232  dmdbr4 10233  dmdi4 10234  dmdbr5 10235  mddmd 10236  mdsl1 10248  cvmd 10251  mdslmd1lem1 10252  mdslmd1lem2 10253  mdslmd1lem3 10254  mdslmd1lem4 10255  mdslmd1 10256  mdslmd3 10259  csmdsym 10261  mdexch 10262  atoml 10309  atabs 10328  sumdmdlem2 10346  dmdbr5at 10349

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051

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