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Theorem iineq1 3671
Description: Equality theorem for restricted existential quantifier. (Contributed by NM, 27-Jun-1998.)
Assertion
Ref Expression
iineq1  |-  ( A  =  B  ->  |^|_ x  e.  A  C  =  |^|_
x  e.  B  C
)
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem iineq1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 raleq 2505 . . 3  |-  ( A  =  B  ->  ( A. x  e.  A  y  e.  C  <->  A. x  e.  B  y  e.  C ) )
21abbidv 2155 . 2  |-  ( A  =  B  ->  { y  |  A. x  e.  A  y  e.  C }  =  { y  |  A. x  e.  B  y  e.  C }
)
3 df-iin 3660 . 2  |-  |^|_ x  e.  A  C  =  { y  |  A. x  e.  A  y  e.  C }
4 df-iin 3660 . 2  |-  |^|_ x  e.  B  C  =  { y  |  A. x  e.  B  y  e.  C }
52, 3, 43eqtr4g 2097 1  |-  ( A  =  B  ->  |^|_ x  e.  A  C  =  |^|_
x  e.  B  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393   {cab 2026   A.wral 2306   |^|_ciin 3658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-iin 3660
This theorem is referenced by:  riin0  3728  iin0r  3922
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