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Theorem iineq12f 31855
Description: Equality deduction for indexed intersections. (Contributed by Giovanni Mascellani, 10-Apr-2018.)
Hypotheses
Ref Expression
iineq12f.1  |-  F/_ x A
iineq12f.2  |-  F/_ x B
Assertion
Ref Expression
iineq12f  |-  ( ( A  =  B  /\  A. x  e.  A  C  =  D )  ->  |^|_ x  e.  A  C  =  |^|_
x  e.  B  D
)

Proof of Theorem iineq12f
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eleq2 2475 . . . . . 6  |-  ( C  =  D  ->  (
y  e.  C  <->  y  e.  D ) )
21ralimi 2797 . . . . 5  |-  ( A. x  e.  A  C  =  D  ->  A. x  e.  A  ( y  e.  C  <->  y  e.  D
) )
3 ralbi 2938 . . . . 5  |-  ( A. x  e.  A  (
y  e.  C  <->  y  e.  D )  ->  ( A. x  e.  A  y  e.  C  <->  A. x  e.  A  y  e.  D ) )
42, 3syl 17 . . . 4  |-  ( A. x  e.  A  C  =  D  ->  ( A. x  e.  A  y  e.  C  <->  A. x  e.  A  y  e.  D )
)
5 iineq12f.1 . . . . 5  |-  F/_ x A
6 iineq12f.2 . . . . 5  |-  F/_ x B
75, 6raleqf 3000 . . . 4  |-  ( A  =  B  ->  ( A. x  e.  A  y  e.  D  <->  A. x  e.  B  y  e.  D ) )
84, 7sylan9bbr 699 . . 3  |-  ( ( A  =  B  /\  A. x  e.  A  C  =  D )  ->  ( A. x  e.  A  y  e.  C  <->  A. x  e.  B  y  e.  D ) )
98abbidv 2538 . 2  |-  ( ( A  =  B  /\  A. x  e.  A  C  =  D )  ->  { y  |  A. x  e.  A  y  e.  C }  =  { y  |  A. x  e.  B  y  e.  D }
)
10 df-iin 4274 . 2  |-  |^|_ x  e.  A  C  =  { y  |  A. x  e.  A  y  e.  C }
11 df-iin 4274 . 2  |-  |^|_ x  e.  B  D  =  { y  |  A. x  e.  B  y  e.  D }
129, 10, 113eqtr4g 2468 1  |-  ( ( A  =  B  /\  A. x  e.  A  C  =  D )  ->  |^|_ x  e.  A  C  =  |^|_
x  e.  B  D
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   {cab 2387   F/_wnfc 2550   A.wral 2754   |^|_ciin 4272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-iin 4274
This theorem is referenced by: (None)
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