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Theorem iineq2i 3924
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 3922 . 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 2612 1  |-  |^|_ x  e.  A  B  =  |^|_
x  e.  A  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   |^|_ciin 3906
This theorem is referenced by:  iinrab  3964  iinin1  3973  rnintintrn  25126  imaiinfv  26759  diaintclN  31248  dibintclN  31357  dihintcl  31534
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-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-ral 2548  df-iin 3908
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