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Theorem bj-seex 34892
Description: Version of seex 4831 with a dv condition replaced by a non-freeness hypothesis (for the sake of illustration). (Contributed by BJ, 27-Apr-2019.)
Hypothesis
Ref Expression
bj-seex.nf  |-  F/_ x B
Assertion
Ref Expression
bj-seex  |-  ( ( R Se  A  /\  B  e.  A )  ->  { x  e.  A  |  x R B }  e.  _V )
Distinct variable groups:    x, A    x, R
Allowed substitution hint:    B( x)

Proof of Theorem bj-seex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-se 4828 . 2  |-  ( R Se  A  <->  A. y  e.  A  { x  e.  A  |  x R y }  e.  _V )
2 bj-seex.nf . . . . . 6  |-  F/_ x B
32nfeq2 2633 . . . . 5  |-  F/ x  y  =  B
4 breq2 4443 . . . . 5  |-  ( y  =  B  ->  (
x R y  <->  x R B ) )
53, 4bj-rabbid 34888 . . . 4  |-  ( y  =  B  ->  { x  e.  A  |  x R y }  =  { x  e.  A  |  x R B }
)
65eleq1d 2523 . . 3  |-  ( y  =  B  ->  ( { x  e.  A  |  x R y }  e.  _V  <->  { x  e.  A  |  x R B }  e.  _V ) )
76rspccva 3206 . 2  |-  ( ( A. y  e.  A  { x  e.  A  |  x R y }  e.  _V  /\  B  e.  A )  ->  { x  e.  A  |  x R B }  e.  _V )
81, 7sylanb 470 1  |-  ( ( R Se  A  /\  B  e.  A )  ->  { x  e.  A  |  x R B }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   F/_wnfc 2602   A.wral 2804   {crab 2808   _Vcvv 3106   class class class wbr 4439   Se wse 4825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-se 4828
This theorem is referenced by: (None)
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