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Theorem 2sb5rf 2001
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
2sb5rf  |-  ( ph  <->  E. z E. 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 2sb5rf
StepHypRef Expression
1 2sb5rf.1 . . 3  |-  ( ph  ->  A. z ph )
21sb5rf 1863 . 2  |-  ( ph  <->  E. z ( z  =  x  /\  [ z  /  x ] ph ) )
3 19.42v 1918 . . . 4  |-  ( E. w ( z  =  x  /\  ( w  =  y  /\  [
w  /  y ] [ z  /  x ] ph ) )  <->  ( z  =  x  /\  E. w
( w  =  y  /\  [ w  / 
y ] [ z  /  x ] ph ) ) )
4 sbcom2 1999 . . . . . . 7  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  [ w  /  y ] [ z  /  x ] ph )
54anbi2i 457 . . . . . 6  |-  ( ( ( z  =  x  /\  w  =  y )  /\  [ z  /  x ] [
w  /  y ]
ph )  <->  ( (
z  =  x  /\  w  =  y )  /\  [ w  /  y ] [ z  /  x ] ph ) )
6 anass 401 . . . . . 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 ) ) )
87exbii 1616 . . . 4  |-  ( E. w ( ( z  =  x  /\  w  =  y )  /\  [ z  /  x ] [ w  /  y ] ph )  <->  E. w
( z  =  x  /\  ( w  =  y  /\  [ w  /  y ] [
z  /  x ] ph ) ) )
9 2sb5rf.2 . . . . . . 7  |-  ( ph  ->  A. w ph )
109hbsbv 1953 . . . . . 6  |-  ( [ z  /  x ] ph  ->  A. w [ z  /  x ] ph )
1110sb5rf 1863 . . . . 5  |-  ( [ z  /  x ] ph 
<->  E. w ( w  =  y  /\  [
w  /  y ] [ z  /  x ] ph ) )
1211anbi2i 457 . . . 4  |-  ( ( z  =  x  /\  [ z  /  x ] ph )  <->  ( z  =  x  /\  E. w
( w  =  y  /\  [ w  / 
y ] [ z  /  x ] ph ) ) )
133, 8, 123bitr4ri 213 . . 3  |-  ( ( z  =  x  /\  [ z  /  x ] ph )  <->  E. w ( ( z  =  x  /\  w  =  y )  /\  [ z  /  x ] [ w  /  y ] ph ) )
1413exbii 1616 . 2  |-  ( E. z ( z  =  x  /\  [ z  /  x ] ph ) 
<->  E. z E. w
( ( z  =  x  /\  w  =  y )  /\  [
z  /  x ] [ w  /  y ] ph ) )
152, 14bitri 184 1  |-  ( ph  <->  E. z E. 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 1362   E.wex 1503   [wsb 1773
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774
This theorem is referenced by: (None)
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