ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2sb6rf Unicode version

Theorem 2sb6rf 1990
Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
Hypotheses
Ref Expression
2sb5rf.1  |-  ( ph  ->  A. z ph )
2sb5rf.2  |-  ( ph  ->  A. w ph )
Assertion
Ref Expression
2sb6rf  |-  ( ph  <->  A. z A. w ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph ) )
Distinct variable groups:    x, y    x, w    y, z    z, w
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem 2sb6rf
StepHypRef Expression
1 2sb5rf.1 . . 3  |-  ( ph  ->  A. z ph )
21sb6rf 1853 . 2  |-  ( ph  <->  A. z ( z  =  x  ->  [ z  /  x ] ph )
)
3 19.21v 1873 . . . 4  |-  ( A. w ( z  =  x  ->  ( w  =  y  ->  [ w  /  y ] [
z  /  x ] ph ) )  <->  ( z  =  x  ->  A. w
( w  =  y  ->  [ w  / 
y ] [ z  /  x ] ph ) ) )
4 sbcom2 1987 . . . . . . 7  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  [ w  /  y ] [ z  /  x ] ph )
54imbi2i 226 . . . . . 6  |-  ( ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph ) 
<->  ( ( z  =  x  /\  w  =  y )  ->  [ w  /  y ] [
z  /  x ] ph ) )
6 impexp 263 . . . . . 6  |-  ( ( ( z  =  x  /\  w  =  y )  ->  [ w  /  y ] [
z  /  x ] ph )  <->  ( z  =  x  ->  ( w  =  y  ->  [ w  /  y ] [
z  /  x ] ph ) ) )
75, 6bitri 184 . . . . 5  |-  ( ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph ) 
<->  ( z  =  x  ->  ( w  =  y  ->  [ w  /  y ] [
z  /  x ] ph ) ) )
87albii 1470 . . . 4  |-  ( A. w ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph )  <->  A. w
( z  =  x  ->  ( w  =  y  ->  [ w  /  y ] [
z  /  x ] ph ) ) )
9 2sb5rf.2 . . . . . . 7  |-  ( ph  ->  A. w ph )
109hbsbv 1941 . . . . . 6  |-  ( [ z  /  x ] ph  ->  A. w [ z  /  x ] ph )
1110sb6rf 1853 . . . . 5  |-  ( [ z  /  x ] ph 
<-> 
A. w ( w  =  y  ->  [ w  /  y ] [
z  /  x ] ph ) )
1211imbi2i 226 . . . 4  |-  ( ( z  =  x  ->  [ z  /  x ] ph )  <->  ( z  =  x  ->  A. w
( w  =  y  ->  [ w  / 
y ] [ z  /  x ] ph ) ) )
133, 8, 123bitr4ri 213 . . 3  |-  ( ( z  =  x  ->  [ z  /  x ] ph )  <->  A. w
( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [
w  /  y ]
ph ) )
1413albii 1470 . 2  |-  ( A. z ( z  =  x  ->  [ z  /  x ] ph )  <->  A. z A. w ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph ) )
152, 14bitri 184 1  |-  ( ph  <->  A. z A. w ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1351   [wsb 1762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator