ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfsb4t Unicode version

Theorem nfsb4t 1906
Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1904). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof rewritten by Jim Kingdon, 9-May-2018.)
Assertion
Ref Expression
nfsb4t  |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )

Proof of Theorem nfsb4t
StepHypRef Expression
1 nfnf1 1452 . . . . 5  |-  F/ z F/ z ph
21nfal 1484 . . . 4  |-  F/ z A. x F/ z
ph
3 nfnae 1626 . . . 4  |-  F/ z  -.  A. z  z  =  y
42, 3nfan 1473 . . 3  |-  F/ z ( A. x F/ z ph  /\  -.  A. z  z  =  y )
5 df-nf 1366 . . . . . 6  |-  ( F/ z ph  <->  A. z
( ph  ->  A. z ph ) )
65albii 1375 . . . . 5  |-  ( A. x F/ z ph  <->  A. x A. z ( ph  ->  A. z ph ) )
7 hbsb4t 1905 . . . . 5  |-  ( A. x A. z ( ph  ->  A. z ph )  ->  ( -.  A. z 
z  =  y  -> 
( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph ) ) )
86, 7sylbi 118 . . . 4  |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph ) ) )
98imp 119 . . 3  |-  ( ( A. x F/ z
ph  /\  -.  A. z 
z  =  y )  ->  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph ) )
104, 9nfd 1432 . 2  |-  ( ( A. x F/ z
ph  /\  -.  A. z 
z  =  y )  ->  F/ z [ y  /  x ] ph )
1110ex 112 1  |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101   A.wal 1257   F/wnf 1365   [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662
This theorem is referenced by:  dvelimdf  1908
  Copyright terms: Public domain W3C validator