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Theorem hbsb4t 1931
Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1930). (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 1474 . . 3  |-  ( A. z ph  ->  A. z A. z ph )
21hbsb4 1930 . 2  |-  ( -. 
A. z  z  =  y  ->  ( [
y  /  x ] A. z ph  ->  A. z [ y  /  x ] A. z ph )
)
3 spsbim 1765 . . . . 5  |-  ( A. x ( ph  ->  A. z ph )  -> 
( [ y  /  x ] ph  ->  [ y  /  x ] A. z ph ) )
43sps 1471 . . . 4  |-  ( A. z A. x ( ph  ->  A. z ph )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] A. z ph ) )
5 ax-4 1441 . . . . . . 7  |-  ( A. z ph  ->  ph )
65sbimi 1688 . . . . . 6  |-  ( [ y  /  x ] A. z ph  ->  [ y  /  x ] ph )
76alimi 1385 . . . . 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 73 . . 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 1384 . 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 1283   [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  nfsb4t  1932
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