Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1468 Structured version   Unicode version

Theorem bnj1468 32983
Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1468.1  |-  ( ps 
->  A. x ps )
bnj1468.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
bnj1468.3  |-  ( y  e.  A  ->  A. x  y  e.  A )
Assertion
Ref Expression
bnj1468  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Distinct variable groups:    y, A    y, V    ph, y    ps, y    x, y
Allowed substitution hints:    ph( x)    ps( x)    A( x)    V( x)

Proof of Theorem bnj1468
StepHypRef Expression
1 sbcco 3354 . 2  |-  ( [. A  /  y ]. [. y  /  x ]. ph  <->  [. A  /  x ]. ph )
2 ax-5 1680 . . 3  |-  ( ps 
->  A. y ps )
3 bnj1468.3 . . . . . . . 8  |-  ( y  e.  A  ->  A. x  y  e.  A )
43nfcii 2619 . . . . . . 7  |-  F/_ x A
54nfeq2 2646 . . . . . 6  |-  F/ x  y  =  A
6 nfsbc1v 3351 . . . . . . 7  |-  F/ x [. y  /  x ]. ph
7 bnj1468.1 . . . . . . . 8  |-  ( ps 
->  A. x ps )
87nfi 1606 . . . . . . 7  |-  F/ x ps
96, 8nfbi 1881 . . . . . 6  |-  F/ x
( [. y  /  x ]. ph  <->  ps )
105, 9nfim 1867 . . . . 5  |-  F/ x
( y  =  A  ->  ( [. y  /  x ]. ph  <->  ps )
)
1110nfri 1822 . . . 4  |-  ( ( y  =  A  -> 
( [. y  /  x ]. ph  <->  ps ) )  ->  A. x ( y  =  A  ->  ( [. y  /  x ]. ph  <->  ps )
) )
12 ax6ev 1721 . . . . 5  |-  E. x  x  =  y
13 eqeq1 2471 . . . . . . 7  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
14 bnj1468.2 . . . . . . 7  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
1513, 14syl6bir 229 . . . . . 6  |-  ( x  =  y  ->  (
y  =  A  -> 
( ph  <->  ps ) ) )
16 sbceq1a 3342 . . . . . . 7  |-  ( x  =  y  ->  ( ph 
<-> 
[. y  /  x ]. ph ) )
1716bibi1d 319 . . . . . 6  |-  ( x  =  y  ->  (
( ph  <->  ps )  <->  ( [. y  /  x ]. ph  <->  ps )
) )
1815, 17sylibd 214 . . . . 5  |-  ( x  =  y  ->  (
y  =  A  -> 
( [. y  /  x ]. ph  <->  ps ) ) )
1912, 18bnj101 32856 . . . 4  |-  E. x
( y  =  A  ->  ( [. y  /  x ]. ph  <->  ps )
)
2011, 19bnj1131 32925 . . 3  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  ps ) )
212, 20bnj1464 32981 . 2  |-  ( A  e.  V  ->  ( [. A  /  y ]. [. y  /  x ]. ph  <->  ps ) )
221, 21syl5bbr 259 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1377    = wceq 1379    e. wcel 1767   [.wsbc 3331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-sbc 3332
This theorem is referenced by:  bnj1463  33190
  Copyright terms: Public domain W3C validator