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Theorem nfsb4t 2131
Description: A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2132). (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-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 sbequ12 1993 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
21sps 1866 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( ph  <->  [ y  /  x ] ph )
)
32drnf2 2073 . . . . . 6  |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z [ y  /  x ] ph ) )
43biimpd 207 . . . . 5  |-  ( A. x  x  =  y  ->  ( F/ z ph  ->  F/ z [ y  /  x ] ph ) )
54spsd 1868 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x F/ z ph  ->  F/ z [ y  /  x ] ph ) )
65impcom 430 . . 3  |-  ( ( A. x F/ z
ph  /\  A. x  x  =  y )  ->  F/ z [ y  /  x ] ph )
76a1d 25 . 2  |-  ( ( A. x F/ z
ph  /\  A. x  x  =  y )  ->  ( -.  A. z 
z  =  y  ->  F/ z [ y  /  x ] ph ) )
8 nfnf1 1900 . . . . 5  |-  F/ z F/ z ph
98nfal 1948 . . . 4  |-  F/ z A. x F/ z
ph
10 nfnae 2059 . . . 4  |-  F/ z  -.  A. x  x  =  y
119, 10nfan 1929 . . 3  |-  F/ z ( A. x F/ z ph  /\  -.  A. x  x  =  y )
12 nfa1 1898 . . . 4  |-  F/ x A. x F/ z ph
13 nfnae 2059 . . . 4  |-  F/ x  -.  A. x  x  =  y
1412, 13nfan 1929 . . 3  |-  F/ x
( A. x F/ z ph  /\  -.  A. x  x  =  y )
15 sp 1860 . . . 4  |-  ( A. x F/ z ph  ->  F/ z ph )
1615adantr 465 . . 3  |-  ( ( A. x F/ z
ph  /\  -.  A. x  x  =  y )  ->  F/ z ph )
17 nfsb2 2101 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
1817adantl 466 . . 3  |-  ( ( A. x F/ z
ph  /\  -.  A. x  x  =  y )  ->  F/ x [ y  /  x ] ph )
191a1i 11 . . 3  |-  ( ( A. x F/ z
ph  /\  -.  A. x  x  =  y )  ->  ( x  =  y  ->  ( ph  <->  [ y  /  x ] ph )
) )
2011, 14, 16, 18, 19dvelimdf 2078 . 2  |-  ( ( A. x F/ z
ph  /\  -.  A. x  x  =  y )  ->  ( -.  A. z 
z  =  y  ->  F/ z [ y  /  x ] ph ) )
217, 20pm2.61dan 791 1  |-  ( A. x F/ z ph  ->  ( -.  A. z  z  =  y  ->  F/ z [ y  /  x ] ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1393   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:  nfsb4  2132  nfsbd  2187
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