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Theorem iineq1 3919
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 2736 . . 3  |-  ( A  =  B  ->  ( A. x  e.  A  y  e.  C  <->  A. x  e.  B  y  e.  C ) )
21abbidv 2397 . 2  |-  ( A  =  B  ->  { y  |  A. x  e.  A  y  e.  C }  =  { y  |  A. x  e.  B  y  e.  C }
)
3 df-iin 3908 . 2  |-  |^|_ x  e.  A  C  =  { y  |  A. x  e.  A  y  e.  C }
4 df-iin 3908 . 2  |-  |^|_ x  e.  B  C  =  { y  |  A. x  e.  B  y  e.  C }
52, 3, 43eqtr4g 2340 1  |-  ( A  =  B  ->  |^|_ x  e.  A  C  =  |^|_
x  e.  B  C
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   |^|_ciin 3906
This theorem is referenced by:  iinrab2  3965  riin0  3975  iin0  4184  xpriindi  4822  cmpfi  17135  ptbasfi  17276  fclsval  17703  taylfval  19738  rnintintrn  25126  polvalN  30094
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-iin 3908
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