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Theorem sbcom2 2077
Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.)
Assertion
Ref Expression
sbcom2  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Distinct variable groups:    x, z    x, w    y, z
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem sbcom2
StepHypRef Expression
1 alcom 1568 . . . . . 6  |-  ( A. z A. x ( x  =  y  ->  (
z  =  w  ->  ph ) )  <->  A. x A. z ( x  =  y  ->  ( z  =  w  ->  ph )
) )
2 bi2.04 352 . . . . . . . . 9  |-  ( ( x  =  y  -> 
( z  =  w  ->  ph ) )  <->  ( z  =  w  ->  ( x  =  y  ->  ph )
) )
32albii 1554 . . . . . . . 8  |-  ( A. x ( x  =  y  ->  ( z  =  w  ->  ph )
)  <->  A. x ( z  =  w  ->  (
x  =  y  ->  ph ) ) )
4 19.21v 2012 . . . . . . . 8  |-  ( A. x ( z  =  w  ->  ( x  =  y  ->  ph )
)  <->  ( z  =  w  ->  A. x
( x  =  y  ->  ph ) ) )
53, 4bitri 242 . . . . . . 7  |-  ( A. x ( x  =  y  ->  ( z  =  w  ->  ph )
)  <->  ( z  =  w  ->  A. x
( x  =  y  ->  ph ) ) )
65albii 1554 . . . . . 6  |-  ( A. z A. x ( x  =  y  ->  (
z  =  w  ->  ph ) )  <->  A. z
( z  =  w  ->  A. x ( x  =  y  ->  ph )
) )
7 19.21v 2012 . . . . . . 7  |-  ( A. z ( x  =  y  ->  ( z  =  w  ->  ph )
)  <->  ( x  =  y  ->  A. z
( z  =  w  ->  ph ) ) )
87albii 1554 . . . . . 6  |-  ( A. x A. z ( x  =  y  ->  (
z  =  w  ->  ph ) )  <->  A. x
( x  =  y  ->  A. z ( z  =  w  ->  ph )
) )
91, 6, 83bitr3i 268 . . . . 5  |-  ( A. z ( z  =  w  ->  A. x
( x  =  y  ->  ph ) )  <->  A. x
( x  =  y  ->  A. z ( z  =  w  ->  ph )
) )
109a1i 12 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  -.  A. z  z  =  w )  ->  ( A. z ( z  =  w  ->  A. x
( x  =  y  ->  ph ) )  <->  A. x
( x  =  y  ->  A. z ( z  =  w  ->  ph )
) ) )
11 sb4b 1947 . . . . 5  |-  ( -. 
A. z  z  =  w  ->  ( [
w  /  z ] [ y  /  x ] ph  <->  A. z ( z  =  w  ->  [ y  /  x ] ph ) ) )
12 sb4b 1947 . . . . . . 7  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
) )
1312imbi2d 309 . . . . . 6  |-  ( -. 
A. x  x  =  y  ->  ( (
z  =  w  ->  [ y  /  x ] ph )  <->  ( z  =  w  ->  A. x
( x  =  y  ->  ph ) ) ) )
1413albidv 2005 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( A. z ( z  =  w  ->  [ y  /  x ] ph )  <->  A. z ( z  =  w  ->  A. x
( x  =  y  ->  ph ) ) ) )
1511, 14sylan9bbr 684 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  -.  A. z  z  =  w )  ->  ( [
w  /  z ] [ y  /  x ] ph  <->  A. z ( z  =  w  ->  A. x
( x  =  y  ->  ph ) ) ) )
16 sb4b 1947 . . . . 5  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] [ w  /  z ] ph  <->  A. x ( x  =  y  ->  [ w  /  z ] ph ) ) )
17 sb4b 1947 . . . . . . 7  |-  ( -. 
A. z  z  =  w  ->  ( [
w  /  z ]
ph 
<-> 
A. z ( z  =  w  ->  ph )
) )
1817imbi2d 309 . . . . . 6  |-  ( -. 
A. z  z  =  w  ->  ( (
x  =  y  ->  [ w  /  z ] ph )  <->  ( x  =  y  ->  A. z
( z  =  w  ->  ph ) ) ) )
1918albidv 2005 . . . . 5  |-  ( -. 
A. z  z  =  w  ->  ( A. x ( x  =  y  ->  [ w  /  z ] ph ) 
<-> 
A. x ( x  =  y  ->  A. z
( z  =  w  ->  ph ) ) ) )
2016, 19sylan9bb 683 . . . 4  |-  ( ( -.  A. x  x  =  y  /\  -.  A. z  z  =  w )  ->  ( [
y  /  x ] [ w  /  z ] ph  <->  A. x ( x  =  y  ->  A. z
( z  =  w  ->  ph ) ) ) )
2110, 15, 203bitr4d 278 . . 3  |-  ( ( -.  A. x  x  =  y  /\  -.  A. z  z  =  w )  ->  ( [
w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
2221ex 425 . 2  |-  ( -. 
A. x  x  =  y  ->  ( -.  A. z  z  =  w  ->  ( [ w  /  z ] [
y  /  x ] ph 
<->  [ y  /  x ] [ w  /  z ] ph ) ) )
23 nfae 1843 . . . 4  |-  F/ z A. x  x  =  y
24 sbequ12 1893 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
2524a4s 1700 . . . 4  |-  ( A. x  x  =  y  ->  ( ph  <->  [ y  /  x ] ph )
)
2623, 25sbbid 1971 . . 3  |-  ( A. x  x  =  y  ->  ( [ w  / 
z ] ph  <->  [ w  /  z ] [
y  /  x ] ph ) )
27 sbequ12 1893 . . . 4  |-  ( x  =  y  ->  ( [ w  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
2827a4s 1700 . . 3  |-  ( A. x  x  =  y  ->  ( [ w  / 
z ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
2926, 28bitr3d 248 . 2  |-  ( A. x  x  =  y  ->  ( [ w  / 
z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
30 sbequ12 1893 . . . 4  |-  ( z  =  w  ->  ( [ y  /  x ] ph  <->  [ w  /  z ] [ y  /  x ] ph ) )
3130a4s 1700 . . 3  |-  ( A. z  z  =  w  ->  ( [ y  /  x ] ph  <->  [ w  /  z ] [
y  /  x ] ph ) )
32 nfae 1843 . . . 4  |-  F/ x A. z  z  =  w
33 sbequ12 1893 . . . . 5  |-  ( z  =  w  ->  ( ph 
<->  [ w  /  z ] ph ) )
3433a4s 1700 . . . 4  |-  ( A. z  z  =  w  ->  ( ph  <->  [ w  /  z ] ph ) )
3532, 34sbbid 1971 . . 3  |-  ( A. z  z  =  w  ->  ( [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
3631, 35bitr3d 248 . 2  |-  ( A. z  z  =  w  ->  ( [ w  / 
z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
3722, 29, 36pm2.61ii 159 1  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532    = wceq 1619   [wsb 1883
This theorem is referenced by:  2sb5rf  2080  2sb6rf  2081  2eu6  2201  cnvopab  5057
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884
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