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Theorem nfsb4t 2194
Description: A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2195). (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 2057 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
21sps 1920 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( ph  <->  [ y  /  x ] ph )
)
32drnf2 2138 . . . . . 6  |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z [ y  /  x ] ph ) )
43biimpd 210 . . . . 5  |-  ( A. x  x  =  y  ->  ( F/ z ph  ->  F/ z [ y  /  x ] ph ) )
54spsd 1922 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x F/ z ph  ->  F/ z [ y  /  x ] ph ) )
65impcom 431 . . 3  |-  ( ( A. x F/ z
ph  /\  A. x  x  =  y )  ->  F/ z [ y  /  x ] ph )
76a1d 26 . 2  |-  ( ( A. x F/ z
ph  /\  A. x  x  =  y )  ->  ( -.  A. z 
z  =  y  ->  F/ z [ y  /  x ] ph ) )
8 nfnf1 1958 . . . . 5  |-  F/ z F/ z ph
98nfal 2007 . . . 4  |-  F/ z A. x F/ z
ph
10 nfnae 2124 . . . 4  |-  F/ z  -.  A. x  x  =  y
119, 10nfan 1988 . . 3  |-  F/ z ( A. x F/ z ph  /\  -.  A. x  x  =  y )
12 nfa1 1956 . . . 4  |-  F/ x A. x F/ z ph
13 nfnae 2124 . . . 4  |-  F/ x  -.  A. x  x  =  y
1412, 13nfan 1988 . . 3  |-  F/ x
( A. x F/ z ph  /\  -.  A. x  x  =  y )
15 sp 1914 . . . 4  |-  ( A. x F/ z ph  ->  F/ z ph )
1615adantr 466 . . 3  |-  ( ( A. x F/ z
ph  /\  -.  A. x  x  =  y )  ->  F/ z ph )
17 nfsb2 2164 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
1817adantl 467 . . 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 2143 . 2  |-  ( ( A. x F/ z
ph  /\  -.  A. x  x  =  y )  ->  ( -.  A. z 
z  =  y  ->  F/ z [ y  /  x ] ph ) )
217, 20pm2.61dan 798 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 187    /\ wa 370   A.wal 1435   F/wnf 1661   [wsb 1790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-sb 1791
This theorem is referenced by:  nfsb4  2195  nfsbd  2248
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