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Theorem sbcralt 3193
Description: Interchange class substitution and restricted quantifier. (Contributed by NM, 1-Mar-2008.) (Revised by David Abernethy, 22-Feb-2010.)
Assertion
Ref Expression
sbcralt  |-  ( ( A  e.  V  /\  F/_ y A )  -> 
( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph )
)
Distinct variable groups:    x, y    x, B
Allowed substitution hints:    ph( x, y)    A( x, y)    B( y)    V( x, y)

Proof of Theorem sbcralt
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbcco 3143 . 2  |-  ( [. A  /  z ]. [. z  /  x ]. A. y  e.  B  ph  <->  [. A  /  x ]. A. y  e.  B  ph )
2 simpl 444 . . 3  |-  ( ( A  e.  V  /\  F/_ y A )  ->  A  e.  V )
3 sbsbc 3125 . . . . 5  |-  ( [ z  /  x ] A. y  e.  B  ph  <->  [. z  /  x ]. A. y  e.  B  ph )
4 nfcv 2540 . . . . . . 7  |-  F/_ x B
5 nfs1v 2155 . . . . . . 7  |-  F/ x [ z  /  x ] ph
64, 5nfral 2719 . . . . . 6  |-  F/ x A. y  e.  B  [ z  /  x ] ph
7 sbequ12 1940 . . . . . . 7  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
87ralbidv 2686 . . . . . 6  |-  ( x  =  z  ->  ( A. y  e.  B  ph  <->  A. y  e.  B  [
z  /  x ] ph ) )
96, 8sbie 2087 . . . . 5  |-  ( [ z  /  x ] A. y  e.  B  ph  <->  A. y  e.  B  [
z  /  x ] ph )
103, 9bitr3i 243 . . . 4  |-  ( [. z  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [ z  /  x ] ph )
11 nfnfc1 2543 . . . . . . 7  |-  F/ y
F/_ y A
12 nfcvd 2541 . . . . . . . 8  |-  ( F/_ y A  ->  F/_ y
z )
13 id 20 . . . . . . . 8  |-  ( F/_ y A  ->  F/_ y A )
1412, 13nfeqd 2554 . . . . . . 7  |-  ( F/_ y A  ->  F/ y  z  =  A )
1511, 14nfan1 1841 . . . . . 6  |-  F/ y ( F/_ y A  /\  z  =  A )
16 dfsbcq2 3124 . . . . . . 7  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
1716adantl 453 . . . . . 6  |-  ( (
F/_ y A  /\  z  =  A )  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
1815, 17ralbid 2684 . . . . 5  |-  ( (
F/_ y A  /\  z  =  A )  ->  ( A. y  e.  B  [ z  /  x ] ph  <->  A. y  e.  B  [. A  /  x ]. ph ) )
1918adantll 695 . . . 4  |-  ( ( ( A  e.  V  /\  F/_ y A )  /\  z  =  A )  ->  ( A. y  e.  B  [
z  /  x ] ph 
<-> 
A. y  e.  B  [. A  /  x ]. ph ) )
2010, 19syl5bb 249 . . 3  |-  ( ( ( A  e.  V  /\  F/_ y A )  /\  z  =  A )  ->  ( [. z  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph ) )
212, 20sbcied 3157 . 2  |-  ( ( A  e.  V  /\  F/_ y A )  -> 
( [. A  /  z ]. [. z  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph )
)
221, 21syl5bbr 251 1  |-  ( ( A  e.  V  /\  F/_ y A )  -> 
( [. A  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [. A  /  x ]. ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649   [wsb 1655    e. wcel 1721   F/_wnfc 2527   A.wral 2666   [.wsbc 3121
This theorem is referenced by:  sbcrext  3194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918  df-sbc 3122
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