ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2sb5 Unicode version

Theorem 2sb5 1875
Description: Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
Assertion
Ref Expression
2sb5  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  E. x E. y
( ( x  =  z  /\  y  =  w )  /\  ph ) )
Distinct variable groups:    x, y, z   
y, w
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem 2sb5
StepHypRef Expression
1 sb5 1783 . 2  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  E. x ( x  =  z  /\  [
w  /  y ]
ph ) )
2 19.42v 1802 . . . 4  |-  ( E. y ( x  =  z  /\  ( y  =  w  /\  ph ) )  <->  ( x  =  z  /\  E. y
( y  =  w  /\  ph ) ) )
3 anass 387 . . . . 5  |-  ( ( ( x  =  z  /\  y  =  w )  /\  ph )  <->  ( x  =  z  /\  ( y  =  w  /\  ph ) ) )
43exbii 1512 . . . 4  |-  ( E. y ( ( x  =  z  /\  y  =  w )  /\  ph ) 
<->  E. y ( x  =  z  /\  (
y  =  w  /\  ph ) ) )
5 sb5 1783 . . . . 5  |-  ( [ w  /  y ]
ph 
<->  E. y ( y  =  w  /\  ph ) )
65anbi2i 438 . . . 4  |-  ( ( x  =  z  /\  [ w  /  y ]
ph )  <->  ( x  =  z  /\  E. y
( y  =  w  /\  ph ) ) )
72, 4, 63bitr4ri 206 . . 3  |-  ( ( x  =  z  /\  [ w  /  y ]
ph )  <->  E. y
( ( x  =  z  /\  y  =  w )  /\  ph ) )
87exbii 1512 . 2  |-  ( E. x ( x  =  z  /\  [ w  /  y ] ph ) 
<->  E. x E. y
( ( x  =  z  /\  y  =  w )  /\  ph ) )
91, 8bitri 177 1  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  E. x E. y
( ( x  =  z  /\  y  =  w )  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102   E.wex 1397   [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  opelopabsbALT  4024
  Copyright terms: Public domain W3C validator