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Theorem sbcralg 3351
Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcralg  |-  ( A  e.  V  ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph )
)
Distinct variable groups:    y, A    x, B    x, y
Allowed substitution hints:    ph( x, y)    A( x)    B( y)    V( x, y)

Proof of Theorem sbcralg
StepHypRef Expression
1 nfcv 2564 . 2  |-  F/_ y A
2 sbcralt 3349 . 2  |-  ( ( A  e.  V  /\  F/_ y A )  -> 
( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph )
)
31, 2mpan2 669 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    e. wcel 1842   F/_wnfc 2550   A.wral 2753   [.wsbc 3276
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-or 368  df-an 369  df-3an 976  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 2758  df-v 3060  df-sbc 3277
This theorem is referenced by:  r19.12sn  4036  r19.12snOLD  4037  bnj538  29110  cdlemkid3N  33932  cdlemkid4  33933  rspsbc2  36305  rspsbc2VD  36665
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