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Theorem 2sb5rf 2069
Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
2sb5rf.1  |-  F/ z
ph
2sb5rf.2  |-  F/ 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  |-  F/ z
ph
21sb5rf 2043 . 2  |-  ( ph  <->  E. z ( z  =  x  /\  [ z  /  x ] ph ) )
3 19.42v 1858 . . . 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 2066 . . . . . . 7  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  [ w  /  y ] [ z  /  x ] ph )
54anbi2i 675 . . . . . 6  |-  ( ( ( z  =  x  /\  w  =  y )  /\  [ z  /  x ] [
w  /  y ]
ph )  <->  ( (
z  =  x  /\  w  =  y )  /\  [ w  /  y ] [ z  /  x ] ph ) )
6 anass 630 . . . . . 6  |-  ( ( ( z  =  x  /\  w  =  y )  /\  [ w  /  y ] [
z  /  x ] ph )  <->  ( z  =  x  /\  ( w  =  y  /\  [
w  /  y ] [ z  /  x ] ph ) ) )
75, 6bitri 240 . . . . 5  |-  ( ( ( z  =  x  /\  w  =  y )  /\  [ z  /  x ] [
w  /  y ]
ph )  <->  ( z  =  x  /\  (
w  =  y  /\  [ w  /  y ] [ z  /  x ] ph ) ) )
87exbii 1572 . . . 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  |-  F/ w ph
109nfsb 2061 . . . . . 6  |-  F/ w [ z  /  x ] ph
1110sb5rf 2043 . . . . 5  |-  ( [ z  /  x ] ph 
<->  E. w ( w  =  y  /\  [
w  /  y ] [ z  /  x ] ph ) )
1211anbi2i 675 . . . 4  |-  ( ( z  =  x  /\  [ z  /  x ] ph )  <->  ( z  =  x  /\  E. w
( w  =  y  /\  [ w  / 
y ] [ z  /  x ] ph ) ) )
133, 8, 123bitr4ri 269 . . 3  |-  ( ( z  =  x  /\  [ z  /  x ] ph )  <->  E. w ( ( z  =  x  /\  w  =  y )  /\  [ z  /  x ] [ w  /  y ] ph ) )
1413exbii 1572 . 2  |-  ( E. z ( z  =  x  /\  [ z  /  x ] ph ) 
<->  E. z E. w
( ( z  =  x  /\  w  =  y )  /\  [
z  /  x ] [ w  /  y ] ph ) )
152, 14bitri 240 1  |-  ( ph  <->  E. z E. w ( ( z  =  x  /\  w  =  y )  /\  [ z  /  x ] [
w  /  y ]
ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531   F/wnf 1534   [wsb 1638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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