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Theorem iineq1 3727
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 2558 . . 3  |-  ( A  =  B  ->  ( A. x  e.  A  y  e.  C  <->  A. x  e.  B  y  e.  C ) )
21abbidv 2202 . 2  |-  ( A  =  B  ->  { y  |  A. x  e.  A  y  e.  C }  =  { y  |  A. x  e.  B  y  e.  C }
)
3 df-iin 3716 . 2  |-  |^|_ x  e.  A  C  =  { y  |  A. x  e.  A  y  e.  C }
4 df-iin 3716 . 2  |-  |^|_ x  e.  B  C  =  { y  |  A. x  e.  B  y  e.  C }
52, 3, 43eqtr4g 2142 1  |-  ( A  =  B  ->  |^|_ x  e.  A  C  =  |^|_
x  e.  B  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    e. wcel 1436   {cab 2071   A.wral 2355   |^|_ciin 3714
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-iin 3716
This theorem is referenced by:  riin0  3784  iin0r  3979
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