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Theorem 2sb6rf 1914
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 1781 . 2  |-  ( ph  <->  A. z ( z  =  x  ->  [ z  /  x ] ph )
)
3 19.21v 1801 . . . 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 1911 . . . . . . 7  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  [ w  /  y ] [ z  /  x ] ph )
54imbi2i 224 . . . . . 6  |-  ( ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph ) 
<->  ( ( z  =  x  /\  w  =  y )  ->  [ w  /  y ] [
z  /  x ] ph ) )
6 impexp 259 . . . . . 6  |-  ( ( ( z  =  x  /\  w  =  y )  ->  [ w  /  y ] [
z  /  x ] ph )  <->  ( z  =  x  ->  ( w  =  y  ->  [ w  /  y ] [
z  /  x ] ph ) ) )
75, 6bitri 182 . . . . 5  |-  ( ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph ) 
<->  ( z  =  x  ->  ( w  =  y  ->  [ w  /  y ] [
z  /  x ] ph ) ) )
87albii 1404 . . . 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 1865 . . . . . 6  |-  ( [ z  /  x ] ph  ->  A. w [ z  /  x ] ph )
1110sb6rf 1781 . . . . 5  |-  ( [ z  /  x ] ph 
<-> 
A. w ( w  =  y  ->  [ w  /  y ] [
z  /  x ] ph ) )
1211imbi2i 224 . . . 4  |-  ( ( z  =  x  ->  [ z  /  x ] ph )  <->  ( z  =  x  ->  A. w
( w  =  y  ->  [ w  / 
y ] [ z  /  x ] ph ) ) )
133, 8, 123bitr4ri 211 . . 3  |-  ( ( z  =  x  ->  [ z  /  x ] ph )  <->  A. w
( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [
w  /  y ]
ph ) )
1413albii 1404 . 2  |-  ( A. z ( z  =  x  ->  [ z  /  x ] ph )  <->  A. z A. w ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph ) )
152, 14bitri 182 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 102    <-> wb 103   A.wal 1287   [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by: (None)
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