ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  hbsb4t Unicode version

Theorem hbsb4t 1993
Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1992). (Contributed by NM, 7-Apr-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Assertion
Ref Expression
hbsb4t  |-  ( A. x A. z ( ph  ->  A. z ph )  ->  ( -.  A. z 
z  =  y  -> 
( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph ) ) )

Proof of Theorem hbsb4t
StepHypRef Expression
1 hba1 1520 . . 3  |-  ( A. z ph  ->  A. z A. z ph )
21hbsb4 1992 . 2  |-  ( -. 
A. z  z  =  y  ->  ( [
y  /  x ] A. z ph  ->  A. z [ y  /  x ] A. z ph )
)
3 spsbim 1823 . . . . 5  |-  ( A. x ( ph  ->  A. z ph )  -> 
( [ y  /  x ] ph  ->  [ y  /  x ] A. z ph ) )
43sps 1517 . . . 4  |-  ( A. z A. x ( ph  ->  A. z ph )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] A. z ph ) )
5 ax-4 1490 . . . . . . 7  |-  ( A. z ph  ->  ph )
65sbimi 1744 . . . . . 6  |-  ( [ y  /  x ] A. z ph  ->  [ y  /  x ] ph )
76alimi 1435 . . . . 5  |-  ( A. z [ y  /  x ] A. z ph  ->  A. z [ y  /  x ] ph )
87a1i 9 . . . 4  |-  ( A. z A. x ( ph  ->  A. z ph )  ->  ( A. z [ y  /  x ] A. z ph  ->  A. z [ y  /  x ] ph ) )
94, 8imim12d 74 . . 3  |-  ( A. z A. x ( ph  ->  A. z ph )  ->  ( ( [ y  /  x ] A. z ph  ->  A. z [ y  /  x ] A. z ph )  ->  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph ) ) )
109a7s 1434 . 2  |-  ( A. x A. z ( ph  ->  A. z ph )  ->  ( ( [ y  /  x ] A. z ph  ->  A. z [ y  /  x ] A. z ph )  ->  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph ) ) )
112, 10syl5 32 1  |-  ( A. x A. z ( ph  ->  A. z ph )  ->  ( -.  A. z 
z  =  y  -> 
( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1333   [wsb 1742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743
This theorem is referenced by:  nfsb4t  1994
  Copyright terms: Public domain W3C validator