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Theorem sbequi 2110
Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Jul-2018.)
Assertion
Ref Expression
sbequi  |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )

Proof of Theorem sbequi
StepHypRef Expression
1 equvini 2083 . . . . 5  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
2 sbequ2 1660 . . . . . . . . 9  |-  ( z  =  x  ->  ( [ x  /  z ] ph  ->  ph ) )
3 sbequ1 1943 . . . . . . . . 9  |-  ( z  =  y  ->  ( ph  ->  [ y  / 
z ] ph )
)
42, 3syl9 68 . . . . . . . 8  |-  ( z  =  x  ->  (
z  =  y  -> 
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) ) )
54equcoms 1693 . . . . . . 7  |-  ( x  =  z  ->  (
z  =  y  -> 
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) ) )
65imp 419 . . . . . 6  |-  ( ( x  =  z  /\  z  =  y )  ->  ( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) )
76eximi 1585 . . . . 5  |-  ( E. z ( x  =  z  /\  z  =  y )  ->  E. z
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) )
81, 7syl 16 . . . 4  |-  ( x  =  y  ->  E. z
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) )
9 nfsb2 2096 . . . . . . 7  |-  ( -. 
A. z  z  =  x  ->  F/ z [ x  /  z ] ph )
109adantr 452 . . . . . 6  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z [ x  /  z ] ph )
11 nfsb2 2096 . . . . . . 7  |-  ( -. 
A. z  z  =  y  ->  F/ z [ y  /  z ] ph )
1211adantl 453 . . . . . 6  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z [ y  /  z ] ph )
1310, 12nfimd 1827 . . . . 5  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) )
141319.9d 1796 . . . 4  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( E. z ( [ x  /  z ] ph  ->  [ y  /  z ] ph )  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )
158, 14syl5 30 . . 3  |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  ( x  =  y  ->  ( [ x  /  z ]
ph  ->  [ y  / 
z ] ph )
) )
1615ex 424 . 2  |-  ( -. 
A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  ( [
x  /  z ]
ph  ->  [ y  / 
z ] ph )
) ) )
17 equtr 1694 . . . 4  |-  ( z  =  x  ->  (
x  =  y  -> 
z  =  y ) )
1817, 4syld 42 . . 3  |-  ( z  =  x  ->  (
x  =  y  -> 
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) ) )
1918sps 1770 . 2  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )
20 equequ2 1698 . . . . . 6  |-  ( x  =  y  ->  (
z  =  x  <->  z  =  y ) )
2120biimprd 215 . . . . 5  |-  ( x  =  y  ->  (
z  =  y  -> 
z  =  x ) )
2221, 4syli 35 . . . 4  |-  ( x  =  y  ->  (
z  =  y  -> 
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) ) )
2322com12 29 . . 3  |-  ( z  =  y  ->  (
x  =  y  -> 
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) ) )
2423sps 1770 . 2  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )
2516, 19, 24pm2.61ii 159 1  |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550   F/wnf 1553   [wsb 1658
This theorem is referenced by:  sbequ  2111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
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