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Theorem 2sb6rf 2057
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
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  |-  F/ z
ph
21sb6rf 2031 . 2  |-  ( ph  <->  A. z ( z  =  x  ->  [ z  /  x ] ph )
)
3 19.21v 1831 . . . 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 2053 . . . . . . 7  |-  ( [ z  /  x ] [ w  /  y ] ph  <->  [ w  /  y ] [ z  /  x ] ph )
54imbi2i 303 . . . . . 6  |-  ( ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph ) 
<->  ( ( z  =  x  /\  w  =  y )  ->  [ w  /  y ] [
z  /  x ] ph ) )
6 impexp 433 . . . . . 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 ) ) )
87albii 1553 . . . 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  |-  F/ w ph
109nfsb 2048 . . . . . 6  |-  F/ w [ z  /  x ] ph
1110sb6rf 2031 . . . . 5  |-  ( [ z  /  x ] ph 
<-> 
A. w ( w  =  y  ->  [ w  /  y ] [
z  /  x ] ph ) )
1211imbi2i 303 . . . 4  |-  ( ( z  =  x  ->  [ z  /  x ] ph )  <->  ( z  =  x  ->  A. w
( w  =  y  ->  [ w  / 
y ] [ z  /  x ] ph ) ) )
133, 8, 123bitr4ri 269 . . 3  |-  ( ( z  =  x  ->  [ z  /  x ] ph )  <->  A. w
( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [
w  /  y ]
ph ) )
1413albii 1553 . 2  |-  ( A. z ( z  =  x  ->  [ z  /  x ] ph )  <->  A. z A. w ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph ) )
152, 14bitri 240 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    <-> wb 176    /\ wa 358   A.wal 1527   F/wnf 1531   [wsb 1629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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