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Theorem nfsb4t 1950
Description: A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1948). (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 1491 . . . . 5  |-  F/ z F/ z ph
21nfal 1523 . . . 4  |-  F/ z A. x F/ z
ph
3 nfnae 1668 . . . 4  |-  F/ z  -.  A. z  z  =  y
42, 3nfan 1512 . . 3  |-  F/ z ( A. x F/ z ph  /\  -.  A. z  z  =  y )
5 df-nf 1405 . . . . . 6  |-  ( F/ z ph  <->  A. z
( ph  ->  A. z ph ) )
65albii 1414 . . . . 5  |-  ( A. x F/ z ph  <->  A. x A. z ( ph  ->  A. z ph ) )
7 hbsb4t 1949 . . . . 5  |-  ( A. x A. z ( ph  ->  A. z ph )  ->  ( -.  A. z 
z  =  y  -> 
( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph ) ) )
86, 7sylbi 120 . . . 4  |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph ) ) )
98imp 123 . . 3  |-  ( ( A. x F/ z
ph  /\  -.  A. z 
z  =  y )  ->  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph ) )
104, 9nfd 1471 . 2  |-  ( ( A. x F/ z
ph  /\  -.  A. z 
z  =  y )  ->  F/ z [ y  /  x ] ph )
1110ex 114 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 103   A.wal 1297   F/wnf 1404   [wsb 1703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704
This theorem is referenced by:  dvelimdf  1952
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