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Theorem sbnf2 2184
Description: Two ways of expressing " x is (effectively) not free in  ph." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Sep-2018.)
Assertion
Ref Expression
sbnf2  |-  ( F/ x ph  <->  A. y A. z ( [ y  /  x ] ph  <->  [ z  /  x ] ph ) )
Distinct variable groups:    x, y,
z    ph, y, z
Allowed substitution hint:    ph( x)

Proof of Theorem sbnf2
StepHypRef Expression
1 nfv 1708 . . . . . 6  |-  F/ y
ph
21sb8e 2169 . . . . 5  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
3 nfv 1708 . . . . . 6  |-  F/ z
ph
43sb8 2168 . . . . 5  |-  ( A. x ph  <->  A. z [ z  /  x ] ph )
52, 4imbi12i 326 . . . 4  |-  ( ( E. x ph  ->  A. x ph )  <->  ( E. y [ y  /  x ] ph  ->  A. z [ z  /  x ] ph ) )
6 nf2 1961 . . . 4  |-  ( F/ x ph  <->  ( E. x ph  ->  A. x ph ) )
7 pm11.53v 1765 . . . 4  |-  ( A. y A. z ( [ y  /  x ] ph  ->  [ z  /  x ] ph )  <->  ( E. y [ y  /  x ] ph  ->  A. z [ z  /  x ] ph ) )
85, 6, 73bitr4i 277 . . 3  |-  ( F/ x ph  <->  A. y A. z ( [ y  /  x ] ph  ->  [ z  /  x ] ph ) )
93sb8e 2169 . . . . . 6  |-  ( E. x ph  <->  E. z [ z  /  x ] ph )
101sb8 2168 . . . . . 6  |-  ( A. x ph  <->  A. y [ y  /  x ] ph )
119, 10imbi12i 326 . . . . 5  |-  ( ( E. x ph  ->  A. x ph )  <->  ( E. z [ z  /  x ] ph  ->  A. y [ y  /  x ] ph ) )
12 pm11.53v 1765 . . . . 5  |-  ( A. z A. y ( [ z  /  x ] ph  ->  [ y  /  x ] ph )  <->  ( E. z [ z  /  x ] ph  ->  A. y [ y  /  x ] ph ) )
1311, 12bitr4i 252 . . . 4  |-  ( ( E. x ph  ->  A. x ph )  <->  A. z A. y ( [ z  /  x ] ph  ->  [ y  /  x ] ph ) )
14 alcom 1846 . . . 4  |-  ( A. z A. y ( [ z  /  x ] ph  ->  [ y  /  x ] ph )  <->  A. y A. z ( [ z  /  x ] ph  ->  [ y  /  x ] ph ) )
156, 13, 143bitri 271 . . 3  |-  ( F/ x ph  <->  A. y A. z ( [ z  /  x ] ph  ->  [ y  /  x ] ph ) )
168, 15anbi12i 697 . 2  |-  ( ( F/ x ph  /\  F/ x ph )  <->  ( A. y A. z ( [ y  /  x ] ph  ->  [ z  /  x ] ph )  /\  A. y A. z ( [ z  /  x ] ph  ->  [ y  /  x ] ph )
) )
17 pm4.24 643 . 2  |-  ( F/ x ph  <->  ( F/ x ph  /\  F/ x ph ) )
18 2albiim 1701 . 2  |-  ( A. y A. z ( [ y  /  x ] ph 
<->  [ z  /  x ] ph )  <->  ( A. y A. z ( [ y  /  x ] ph  ->  [ z  /  x ] ph )  /\  A. y A. z ( [ z  /  x ] ph  ->  [ y  /  x ] ph )
) )
1916, 17, 183bitr4i 277 1  |-  ( F/ x ph  <->  A. y A. z ( [ y  /  x ] ph  <->  [ z  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1393   E.wex 1613   F/wnf 1617   [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-an 371  df-ex 1614  df-nf 1618  df-sb 1741
This theorem is referenced by:  sbnfc2  3859  nfnid  4685
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