Users' Mathboxes Mathbox for Wolf Lammen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  wl-sbcom2d-lem1 Structured version   Unicode version

Theorem wl-sbcom2d-lem1 30214
Description: Lemma used to prove wl-sbcom2d 30216. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.)
Assertion
Ref Expression
wl-sbcom2d-lem1  |-  ( ( u  =  y  /\  v  =  w )  ->  ( -.  A. x  x  =  w  ->  ( [ u  /  x ] [ v  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) ) )
Distinct variable groups:    v, u, x    y, u, v    w, u, v    z, u, v    ph, u, v
Allowed substitution hints:    ph( x, y, z, w)

Proof of Theorem wl-sbcom2d-lem1
StepHypRef Expression
1 nfna1 1904 . . . . . 6  |-  F/ x  -.  A. x  x  =  w
2 nfeqf2 2042 . . . . . 6  |-  ( -. 
A. x  x  =  w  ->  F/ x  v  =  w )
31, 2nfan1 1928 . . . . 5  |-  F/ x
( -.  A. x  x  =  w  /\  v  =  w )
4 sbequ 2118 . . . . . 6  |-  ( v  =  w  ->  ( [ v  /  z ] ph  <->  [ w  /  z ] ph ) )
54adantl 466 . . . . 5  |-  ( ( -.  A. x  x  =  w  /\  v  =  w )  ->  ( [ v  /  z ] ph  <->  [ w  /  z ] ph ) )
63, 5sbbid 2145 . . . 4  |-  ( ( -.  A. x  x  =  w  /\  v  =  w )  ->  ( [ u  /  x ] [ v  /  z ] ph  <->  [ u  /  x ] [ w  /  z ] ph ) )
76ancoms 453 . . 3  |-  ( ( v  =  w  /\  -.  A. x  x  =  w )  ->  ( [ u  /  x ] [ v  /  z ] ph  <->  [ u  /  x ] [ w  /  z ] ph ) )
8 sbequ 2118 . . 3  |-  ( u  =  y  ->  ( [ u  /  x ] [ w  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
97, 8sylan9bbr 700 . 2  |-  ( ( u  =  y  /\  ( v  =  w  /\  -.  A. x  x  =  w )
)  ->  ( [
u  /  x ] [ v  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) )
109expr 615 1  |-  ( ( u  =  y  /\  v  =  w )  ->  ( -.  A. x  x  =  w  ->  ( [ u  /  x ] [ v  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1393   [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-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:  wl-sbcom2d  30216
  Copyright terms: Public domain W3C validator