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Theorem sbequi 2012
Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbequi  |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )

Proof of Theorem sbequi
StepHypRef Expression
1 hbsb2 2010 . . . . . 6  |-  ( -. 
A. z  z  =  x  ->  ( [
x  /  z ]
ph  ->  A. z [ x  /  z ] ph ) )
2 equvini 1940 . . . . . . . 8  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
3 stdpc7 1870 . . . . . . . . . 10  |-  ( x  =  z  ->  ( [ x  /  z ] ph  ->  ph ) )
4 sbequ1 1871 . . . . . . . . . 10  |-  ( z  =  y  ->  ( ph  ->  [ y  / 
z ] ph )
)
53, 4sylan9 638 . . . . . . . . 9  |-  ( ( x  =  z  /\  z  =  y )  ->  ( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) )
65eximi 1566 . . . . . . . 8  |-  ( E. z ( x  =  z  /\  z  =  y )  ->  E. z
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) )
72, 6syl 15 . . . . . . 7  |-  ( x  =  y  ->  E. z
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) )
8 19.35 1590 . . . . . . 7  |-  ( E. z ( [ x  /  z ] ph  ->  [ y  /  z ] ph )  <->  ( A. z [ x  /  z ] ph  ->  E. z [ y  /  z ] ph ) )
97, 8sylib 188 . . . . . 6  |-  ( x  =  y  ->  ( A. z [ x  / 
z ] ph  ->  E. z [ y  / 
z ] ph )
)
101, 9sylan9 638 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  x  =  y )  -> 
( [ x  / 
z ] ph  ->  E. z [ y  / 
z ] ph )
)
11 nfsb2 2011 . . . . . 6  |-  ( -. 
A. z  z  =  y  ->  F/ z [ y  /  z ] ph )
121119.9d 1796 . . . . 5  |-  ( -. 
A. z  z  =  y  ->  ( E. z [ y  /  z ] ph  ->  [ y  /  z ] ph ) )
1310, 12syl9 66 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  x  =  y )  -> 
( -.  A. z 
z  =  y  -> 
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) ) )
1413ex 423 . . 3  |-  ( -. 
A. z  z  =  x  ->  ( x  =  y  ->  ( -. 
A. z  z  =  y  ->  ( [
x  /  z ]
ph  ->  [ y  / 
z ] ph )
) ) )
1514com23 72 . 2  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  ( [
x  /  z ]
ph  ->  [ y  / 
z ] ph )
) ) )
16 sbequ2 1640 . . . . . 6  |-  ( z  =  x  ->  ( [ x  /  z ] ph  ->  ph ) )
1716sps 1751 . . . . 5  |-  ( A. z  z  =  x  ->  ( [ x  / 
z ] ph  ->  ph ) )
1817adantr 451 . . . 4  |-  ( ( A. z  z  =  x  /\  x  =  y )  ->  ( [ x  /  z ] ph  ->  ph ) )
19 sbequ1 1871 . . . . 5  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
20 drsb1 1975 . . . . . 6  |-  ( A. z  z  =  x  ->  ( [ y  / 
z ] ph  <->  [ y  /  x ] ph )
)
2120biimprd 214 . . . . 5  |-  ( A. z  z  =  x  ->  ( [ y  /  x ] ph  ->  [ y  /  z ] ph ) )
2219, 21sylan9r 639 . . . 4  |-  ( ( A. z  z  =  x  /\  x  =  y )  ->  ( ph  ->  [ y  / 
z ] ph )
)
2318, 22syld 40 . . 3  |-  ( ( A. z  z  =  x  /\  x  =  y )  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )
2423ex 423 . 2  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )
25 drsb1 1975 . . . . . 6  |-  ( A. z  z  =  y  ->  ( [ x  / 
z ] ph  <->  [ x  /  y ] ph ) )
2625biimpd 198 . . . . 5  |-  ( A. z  z  =  y  ->  ( [ x  / 
z ] ph  ->  [ x  /  y ]
ph ) )
27 stdpc7 1870 . . . . 5  |-  ( x  =  y  ->  ( [ x  /  y ] ph  ->  ph ) )
2826, 27sylan9 638 . . . 4  |-  ( ( A. z  z  =  y  /\  x  =  y )  ->  ( [ x  /  z ] ph  ->  ph ) )
294sps 1751 . . . . 5  |-  ( A. z  z  =  y  ->  ( ph  ->  [ y  /  z ] ph ) )
3029adantr 451 . . . 4  |-  ( ( A. z  z  =  y  /\  x  =  y )  ->  ( ph  ->  [ y  / 
z ] ph )
)
3128, 30syld 40 . . 3  |-  ( ( A. z  z  =  y  /\  x  =  y )  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )
3231ex 423 . 2  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )
3315, 24, 32pm2.61ii 157 1  |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1530   E.wex 1531   [wsb 1638
This theorem is referenced by:  sbequ  2013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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