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Theorem iineq2i 3676
Description: Equality inference for indexed intersection. (Contributed by NM, 22-Oct-2003.)
Hypothesis
Ref Expression
iuneq2i.1  |-  ( x  e.  A  ->  B  =  C )
Assertion
Ref Expression
iineq2i  |-  |^|_ x  e.  A  B  =  |^|_
x  e.  A  C

Proof of Theorem iineq2i
StepHypRef Expression
1 iineq2 3674 . 2  |-  ( A. x  e.  A  B  =  C  ->  |^|_ x  e.  A  B  =  |^|_
x  e.  A  C
)
2 iuneq2i.1 . 2  |-  ( x  e.  A  ->  B  =  C )
31, 2mprg 2378 1  |-  |^|_ x  e.  A  B  =  |^|_
x  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    e. wcel 1393   |^|_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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  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-ral 2311  df-iin 3660
This theorem is referenced by:  iinrabm  3719  iinin1m  3726
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