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Theorem sbcom2 2190
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 1750 . 2  |-  E. u  u  =  y
2 ax6ev 1750 . 2  |-  E. v 
v  =  w
3 2sb6 2189 . . . . . . . . . 10  |-  ( [ v  /  z ] [ u  /  x ] ph  <->  A. z A. x
( ( z  =  v  /\  x  =  u )  ->  ph )
)
4 alcom 1846 . . . . . . . . . 10  |-  ( A. z A. x ( ( z  =  v  /\  x  =  u )  ->  ph )  <->  A. x A. z ( ( z  =  v  /\  x  =  u )  ->  ph )
)
5 ancomst 452 . . . . . . . . . . 11  |-  ( ( ( z  =  v  /\  x  =  u )  ->  ph )  <->  ( (
x  =  u  /\  z  =  v )  ->  ph ) )
652albii 1642 . . . . . . . . . 10  |-  ( A. x A. z ( ( z  =  v  /\  x  =  u )  ->  ph )  <->  A. x A. z ( ( x  =  u  /\  z  =  v )  ->  ph ) )
73, 4, 63bitri 271 . . . . . . . . 9  |-  ( [ v  /  z ] [ u  /  x ] ph  <->  A. x A. z
( ( x  =  u  /\  z  =  v )  ->  ph )
)
8 2sb6 2189 . . . . . . . . 9  |-  ( [ u  /  x ] [ v  /  z ] ph  <->  A. x A. z
( ( x  =  u  /\  z  =  v )  ->  ph )
)
97, 8bitr4i 252 . . . . . . . 8  |-  ( [ v  /  z ] [ u  /  x ] ph  <->  [ u  /  x ] [ v  /  z ] ph )
10 nfv 1708 . . . . . . . . 9  |-  F/ z  u  =  y
11 sbequ 2118 . . . . . . . . 9  |-  ( u  =  y  ->  ( [ u  /  x ] ph  <->  [ y  /  x ] ph ) )
1210, 11sbbid 2145 . . . . . . . 8  |-  ( u  =  y  ->  ( [ v  /  z ] [ u  /  x ] ph  <->  [ v  /  z ] [ y  /  x ] ph ) )
139, 12syl5bbr 259 . . . . . . 7  |-  ( u  =  y  ->  ( [ u  /  x ] [ v  /  z ] ph  <->  [ v  /  z ] [ y  /  x ] ph ) )
14 sbequ 2118 . . . . . . 7  |-  ( v  =  w  ->  ( [ v  /  z ] [ y  /  x ] ph  <->  [ w  /  z ] [ y  /  x ] ph ) )
1513, 14sylan9bb 699 . . . . . 6  |-  ( ( u  =  y  /\  v  =  w )  ->  ( [ u  /  x ] [ v  / 
z ] ph  <->  [ w  /  z ] [
y  /  x ] ph ) )
16 nfv 1708 . . . . . . . 8  |-  F/ x  v  =  w
17 sbequ 2118 . . . . . . . 8  |-  ( v  =  w  ->  ( [ v  /  z ] ph  <->  [ w  /  z ] ph ) )
1816, 17sbbid 2145 . . . . . . 7  |-  ( v  =  w  ->  ( [ u  /  x ] [ v  /  z ] ph  <->  [ u  /  x ] [ w  /  z ] ph ) )
19 sbequ 2118 . . . . . . 7  |-  ( u  =  y  ->  ( [ u  /  x ] [ w  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
2018, 19sylan9bbr 700 . . . . . 6  |-  ( ( u  =  y  /\  v  =  w )  ->  ( [ u  /  x ] [ v  / 
z ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
2115, 20bitr3d 255 . . . . 5  |-  ( ( u  =  y  /\  v  =  w )  ->  ( [ w  / 
z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
2221ex 434 . . . 4  |-  ( u  =  y  ->  (
v  =  w  -> 
( [ w  / 
z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) ) )
2322exlimdv 1725 . . 3  |-  ( u  =  y  ->  ( E. v  v  =  w  ->  ( [ w  /  z ] [
y  /  x ] ph 
<->  [ y  /  x ] [ w  /  z ] ph ) ) )
2423exlimiv 1723 . 2  |-  ( E. u  u  =  y  ->  ( E. v 
v  =  w  -> 
( [ w  / 
z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) ) )
251, 2, 24mp2 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 1393   E.wex 1613   [wsb 1740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1614  df-nf 1618  df-sb 1741
This theorem is referenced by:  sbco4lem  2210  sbco4  2211  2mo  2373  2moOLD  2374  2eu6OLD  2384  cnvopab  5414
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