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Theorem iineq12f 29112
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 2522 . . . . . 6  |-  ( C  =  D  ->  (
y  e.  C  <->  y  e.  D ) )
21ralimi 2809 . . . . 5  |-  ( A. x  e.  A  C  =  D  ->  A. x  e.  A  ( y  e.  C  <->  y  e.  D
) )
3 ralbi 2946 . . . . 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 16 . . . 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 3006 . . . 4  |-  ( A  =  B  ->  ( A. x  e.  A  y  e.  D  <->  A. x  e.  B  y  e.  D ) )
84, 7sylan9bbr 700 . . 3  |-  ( ( A  =  B  /\  A. x  e.  A  C  =  D )  ->  ( A. x  e.  A  y  e.  C  <->  A. x  e.  B  y  e.  D ) )
98abbidv 2585 . 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 4269 . 2  |-  |^|_ x  e.  A  C  =  { y  |  A. x  e.  A  y  e.  C }
11 df-iin 4269 . 2  |-  |^|_ x  e.  B  D  =  { y  |  A. x  e.  B  y  e.  D }
129, 10, 113eqtr4g 2516 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 369    = wceq 1370    e. wcel 1758   {cab 2436   F/_wnfc 2597   A.wral 2793   |^|_ciin 4267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ral 2798  df-iin 4269
This theorem is referenced by: (None)
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