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Theorem nfsb4t 2229
Description: A variable not free remains so after substitution with a distinct variable (closed form of nfsb4 2230). (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 2094 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
21sps 1954 . . . . . . 7  |-  ( A. x  x  =  y  ->  ( ph  <->  [ y  /  x ] ph )
)
32drnf2 2175 . . . . . 6  |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z [ y  /  x ] ph ) )
43biimpd 212 . . . . 5  |-  ( A. x  x  =  y  ->  ( F/ z ph  ->  F/ z [ y  /  x ] ph ) )
54spsd 1956 . . . 4  |-  ( A. x  x  =  y  ->  ( A. x F/ z ph  ->  F/ z [ y  /  x ] ph ) )
65impcom 436 . . 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 1992 . . . . 5  |-  F/ z F/ z ph
98nfal 2041 . . . 4  |-  F/ z A. x F/ z
ph
10 nfnae 2163 . . . 4  |-  F/ z  -.  A. x  x  =  y
119, 10nfan 2022 . . 3  |-  F/ z ( A. x F/ z ph  /\  -.  A. x  x  =  y )
12 nfa1 1990 . . . 4  |-  F/ x A. x F/ z ph
13 nfnae 2163 . . . 4  |-  F/ x  -.  A. x  x  =  y
1412, 13nfan 2022 . . 3  |-  F/ x
( A. x F/ z ph  /\  -.  A. x  x  =  y )
15 sp 1948 . . . 4  |-  ( A. x F/ z ph  ->  F/ z ph )
1615adantr 471 . . 3  |-  ( ( A. x F/ z
ph  /\  -.  A. x  x  =  y )  ->  F/ z ph )
17 nfsb2 2201 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
1817adantl 472 . . 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 2180 . 2  |-  ( ( A. x F/ z
ph  /\  -.  A. x  x  =  y )  ->  ( -.  A. z 
z  =  y  ->  F/ z [ y  /  x ] ph ) )
217, 20pm2.61dan 805 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 189    /\ wa 375   A.wal 1453   F/wnf 1678   [wsb 1808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679  df-sb 1809
This theorem is referenced by:  nfsb4  2230  nfsbd  2282
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