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Theorem sbcom2 2151
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.) (Proof shortened by Wolf Lammen, 24-Sep-2018.)
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
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax6ev 1710 . 2  |-  E. u  u  =  y
2 ax6ev 1710 . 2  |-  E. v 
v  =  w
3 2sb6 2150 . . . . . . . . . 10  |-  ( [ v  /  z ] [ u  /  x ] ph  <->  A. z A. x
( ( z  =  v  /\  x  =  u )  ->  ph )
)
4 alcom 1783 . . . . . . . . . 10  |-  ( A. z A. x ( ( z  =  v  /\  x  =  u )  ->  ph )  <->  A. x A. z ( ( z  =  v  /\  x  =  u )  ->  ph )
)
5 ancom 450 . . . . . . . . . . . 12  |-  ( ( z  =  v  /\  x  =  u )  <->  ( x  =  u  /\  z  =  v )
)
65imbi1i 325 . . . . . . . . . . 11  |-  ( ( ( z  =  v  /\  x  =  u )  ->  ph )  <->  ( (
x  =  u  /\  z  =  v )  ->  ph ) )
762albii 1611 . . . . . . . . . 10  |-  ( A. x A. z ( ( z  =  v  /\  x  =  u )  ->  ph )  <->  A. x A. z ( ( x  =  u  /\  z  =  v )  ->  ph ) )
83, 4, 73bitri 271 . . . . . . . . 9  |-  ( [ v  /  z ] [ u  /  x ] ph  <->  A. x A. z
( ( x  =  u  /\  z  =  v )  ->  ph )
)
9 2sb6 2150 . . . . . . . . 9  |-  ( [ u  /  x ] [ v  /  z ] ph  <->  A. x A. z
( ( x  =  u  /\  z  =  v )  ->  ph )
)
108, 9bitr4i 252 . . . . . . . 8  |-  ( [ v  /  z ] [ u  /  x ] ph  <->  [ u  /  x ] [ v  /  z ] ph )
11 nfv 1673 . . . . . . . . 9  |-  F/ z  u  =  y
12 sbequ 2067 . . . . . . . . 9  |-  ( u  =  y  ->  ( [ u  /  x ] ph  <->  [ y  /  x ] ph ) )
1311, 12sbbid 2095 . . . . . . . 8  |-  ( u  =  y  ->  ( [ v  /  z ] [ u  /  x ] ph  <->  [ v  /  z ] [ y  /  x ] ph ) )
1410, 13syl5bbr 259 . . . . . . 7  |-  ( u  =  y  ->  ( [ u  /  x ] [ v  /  z ] ph  <->  [ v  /  z ] [ y  /  x ] ph ) )
15 sbequ 2067 . . . . . . 7  |-  ( v  =  w  ->  ( [ v  /  z ] [ y  /  x ] ph  <->  [ w  /  z ] [ y  /  x ] ph ) )
1614, 15sylan9bb 699 . . . . . 6  |-  ( ( u  =  y  /\  v  =  w )  ->  ( [ u  /  x ] [ v  / 
z ] ph  <->  [ w  /  z ] [
y  /  x ] ph ) )
17 nfv 1673 . . . . . . . 8  |-  F/ x  v  =  w
18 sbequ 2067 . . . . . . . 8  |-  ( v  =  w  ->  ( [ v  /  z ] ph  <->  [ w  /  z ] ph ) )
1917, 18sbbid 2095 . . . . . . 7  |-  ( v  =  w  ->  ( [ u  /  x ] [ v  /  z ] ph  <->  [ u  /  x ] [ w  /  z ] ph ) )
20 sbequ 2067 . . . . . . 7  |-  ( u  =  y  ->  ( [ u  /  x ] [ w  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
2119, 20sylan9bbr 700 . . . . . 6  |-  ( ( u  =  y  /\  v  =  w )  ->  ( [ u  /  x ] [ v  / 
z ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
2216, 21bitr3d 255 . . . . 5  |-  ( ( u  =  y  /\  v  =  w )  ->  ( [ w  / 
z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
2322ex 434 . . . 4  |-  ( u  =  y  ->  (
v  =  w  -> 
( [ w  / 
z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) ) )
2423exlimdv 1690 . . 3  |-  ( u  =  y  ->  ( E. v  v  =  w  ->  ( [ w  /  z ] [
y  /  x ] ph 
<->  [ y  /  x ] [ w  /  z ] ph ) ) )
2524exlimiv 1688 . 2  |-  ( E. u  u  =  y  ->  ( E. v 
v  =  w  -> 
( [ w  / 
z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) ) )
261, 2, 25mp2 9 1  |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367   E.wex 1586   [wsb 1700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701
This theorem is referenced by:  2sb5rfOLD  2161  2sb6rfOLD  2162  sbco4lem  2178  sbco4  2179  2mo  2360  2moOLD  2361  2eu6OLD  2372  cnvopab  5238
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