MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcralt Structured version   Visualization version   Unicode version

Theorem sbcralt 3351
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 3301 . 2  |-  ( [. A  /  z ]. [. z  /  x ]. A. y  e.  B  ph  <->  [. A  /  x ]. A. y  e.  B  ph )
2 simpl 463 . . 3  |-  ( ( A  e.  V  /\  F/_ y A )  ->  A  e.  V )
3 sbsbc 3282 . . . . 5  |-  ( [ z  /  x ] A. y  e.  B  ph  <->  [. z  /  x ]. A. y  e.  B  ph )
4 nfcv 2602 . . . . . . 7  |-  F/_ x B
5 nfs1v 2276 . . . . . . 7  |-  F/ x [ z  /  x ] ph
64, 5nfral 2785 . . . . . 6  |-  F/ x A. y  e.  B  [ z  /  x ] ph
7 sbequ12 2093 . . . . . . 7  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
87ralbidv 2838 . . . . . 6  |-  ( x  =  z  ->  ( A. y  e.  B  ph  <->  A. y  e.  B  [
z  /  x ] ph ) )
96, 8sbie 2247 . . . . 5  |-  ( [ z  /  x ] A. y  e.  B  ph  <->  A. y  e.  B  [
z  /  x ] ph )
103, 9bitr3i 259 . . . 4  |-  ( [. z  /  x ]. A. y  e.  B  ph  <->  A. y  e.  B  [ z  /  x ] ph )
11 nfnfc1 2605 . . . . . . 7  |-  F/ y
F/_ y A
12 nfcvd 2603 . . . . . . . 8  |-  ( F/_ y A  ->  F/_ y
z )
13 id 22 . . . . . . . 8  |-  ( F/_ y A  ->  F/_ y A )
1412, 13nfeqd 2609 . . . . . . 7  |-  ( F/_ y A  ->  F/ y  z  =  A )
1511, 14nfan1 2020 . . . . . 6  |-  F/ y ( F/_ y A  /\  z  =  A )
16 dfsbcq2 3281 . . . . . . 7  |-  ( z  =  A  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
1716adantl 472 . . . . . 6  |-  ( (
F/_ y A  /\  z  =  A )  ->  ( [ z  /  x ] ph  <->  [. A  /  x ]. ph ) )
1815, 17ralbid 2833 . . . . 5  |-  ( (
F/_ y A  /\  z  =  A )  ->  ( A. y  e.  B  [ z  /  x ] ph  <->  A. y  e.  B  [. A  /  x ]. ph ) )
1918adantll 725 . . . 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 265 . . 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 3315 . 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 267 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454   [wsb 1807    e. wcel 1897   F/_wnfc 2589   A.wral 2748   [.wsbc 3278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ral 2753  df-v 3058  df-sbc 3279
This theorem is referenced by:  sbcrext  3352  sbcralg  3353
  Copyright terms: Public domain W3C validator