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Theorem sbequi 1767
Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.)
Assertion
Ref Expression
sbequi  |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )

Proof of Theorem sbequi
StepHypRef Expression
1 nfsb2or 1765 . . . 4  |-  ( A. z  z  =  x  \/  F/ z [ x  /  z ] ph )
2 nfr 1456 . . . . . 6  |-  ( F/ z [ x  / 
z ] ph  ->  ( [ x  /  z ] ph  ->  A. z [ x  /  z ] ph ) )
3 equvini 1688 . . . . . . 7  |-  ( x  =  y  ->  E. z
( x  =  z  /\  z  =  y ) )
4 stdpc7 1700 . . . . . . . . 9  |-  ( x  =  z  ->  ( [ x  /  z ] ph  ->  ph ) )
5 sbequ1 1698 . . . . . . . . 9  |-  ( z  =  y  ->  ( ph  ->  [ y  / 
z ] ph )
)
64, 5sylan9 401 . . . . . . . 8  |-  ( ( x  =  z  /\  z  =  y )  ->  ( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) )
76eximi 1536 . . . . . . 7  |-  ( E. z ( x  =  z  /\  z  =  y )  ->  E. z
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) )
8 19.35-1 1560 . . . . . . 7  |-  ( E. z ( [ x  /  z ] ph  ->  [ y  /  z ] ph )  ->  ( A. z [ x  / 
z ] ph  ->  E. z [ y  / 
z ] ph )
)
93, 7, 83syl 17 . . . . . 6  |-  ( x  =  y  ->  ( A. z [ x  / 
z ] ph  ->  E. z [ y  / 
z ] ph )
)
102, 9syl9 71 . . . . 5  |-  ( F/ z [ x  / 
z ] ph  ->  ( x  =  y  -> 
( [ x  / 
z ] ph  ->  E. z [ y  / 
z ] ph )
) )
1110orim2i 713 . . . 4  |-  ( ( A. z  z  =  x  \/  F/ z [ x  /  z ] ph )  ->  ( A. z  z  =  x  \/  ( x  =  y  ->  ( [ x  /  z ]
ph  ->  E. z [ y  /  z ] ph ) ) ) )
121, 11ax-mp 7 . . 3  |-  ( A. z  z  =  x  \/  ( x  =  y  ->  ( [ x  /  z ] ph  ->  E. z [ y  /  z ] ph ) ) )
13 nfsb2or 1765 . . . . 5  |-  ( A. z  z  =  y  \/  F/ z [ y  /  z ] ph )
14 19.9t 1578 . . . . . . 7  |-  ( F/ z [ y  / 
z ] ph  ->  ( E. z [ y  /  z ] ph  <->  [ y  /  z ]
ph ) )
1514biimpd 142 . . . . . 6  |-  ( F/ z [ y  / 
z ] ph  ->  ( E. z [ y  /  z ] ph  ->  [ y  /  z ] ph ) )
1615orim2i 713 . . . . 5  |-  ( ( A. z  z  =  y  \/  F/ z [ y  /  z ] ph )  ->  ( A. z  z  =  y  \/  ( E. z [ y  /  z ] ph  ->  [ y  /  z ] ph ) ) )
1713, 16ax-mp 7 . . . 4  |-  ( A. z  z  =  y  \/  ( E. z [ y  /  z ]
ph  ->  [ y  / 
z ] ph )
)
18 ax-1 5 . . . . 5  |-  ( ( E. z [ y  /  z ] ph  ->  [ y  /  z ] ph )  ->  (
x  =  y  -> 
( E. z [ y  /  z ]
ph  ->  [ y  / 
z ] ph )
) )
1918orim2i 713 . . . 4  |-  ( ( A. z  z  =  y  \/  ( E. z [ y  / 
z ] ph  ->  [ y  /  z ]
ph ) )  -> 
( A. z  z  =  y  \/  (
x  =  y  -> 
( E. z [ y  /  z ]
ph  ->  [ y  / 
z ] ph )
) ) )
2017, 19ax-mp 7 . . 3  |-  ( A. z  z  =  y  \/  ( x  =  y  ->  ( E. z [ y  /  z ] ph  ->  [ y  /  z ] ph ) ) )
2112, 20sbequilem 1766 . 2  |-  ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  (
x  =  y  -> 
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) ) ) )
22 sbequ2 1699 . . . . . . 7  |-  ( z  =  x  ->  ( [ x  /  z ] ph  ->  ph ) )
2322sps 1475 . . . . . 6  |-  ( A. z  z  =  x  ->  ( [ x  / 
z ] ph  ->  ph ) )
2423adantr 270 . . . . 5  |-  ( ( A. z  z  =  x  /\  x  =  y )  ->  ( [ x  /  z ] ph  ->  ph ) )
25 sbequ1 1698 . . . . . 6  |-  ( x  =  y  ->  ( ph  ->  [ y  /  x ] ph ) )
26 drsb1 1727 . . . . . . . 8  |-  ( A. x  x  =  z  ->  ( [ y  /  x ] ph  <->  [ y  /  z ] ph ) )
2726biimpd 142 . . . . . . 7  |-  ( A. x  x  =  z  ->  ( [ y  /  x ] ph  ->  [ y  /  z ] ph ) )
2827alequcoms 1454 . . . . . 6  |-  ( A. z  z  =  x  ->  ( [ y  /  x ] ph  ->  [ y  /  z ] ph ) )
2925, 28sylan9r 402 . . . . 5  |-  ( ( A. z  z  =  x  /\  x  =  y )  ->  ( ph  ->  [ y  / 
z ] ph )
)
3024, 29syld 44 . . . 4  |-  ( ( A. z  z  =  x  /\  x  =  y )  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )
3130ex 113 . . 3  |-  ( A. z  z  =  x  ->  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )
32 drsb1 1727 . . . . . . . . 9  |-  ( A. z  z  =  y  ->  ( [ x  / 
z ] ph  <->  [ x  /  y ] ph ) )
3332biimpd 142 . . . . . . . 8  |-  ( A. z  z  =  y  ->  ( [ x  / 
z ] ph  ->  [ x  /  y ]
ph ) )
34 stdpc7 1700 . . . . . . . 8  |-  ( x  =  y  ->  ( [ x  /  y ] ph  ->  ph ) )
3533, 34sylan9 401 . . . . . . 7  |-  ( ( A. z  z  =  y  /\  x  =  y )  ->  ( [ x  /  z ] ph  ->  ph ) )
365sps 1475 . . . . . . . 8  |-  ( A. z  z  =  y  ->  ( ph  ->  [ y  /  z ] ph ) )
3736adantr 270 . . . . . . 7  |-  ( ( A. z  z  =  y  /\  x  =  y )  ->  ( ph  ->  [ y  / 
z ] ph )
)
3835, 37syld 44 . . . . . 6  |-  ( ( A. z  z  =  y  /\  x  =  y )  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )
3938ex 113 . . . . 5  |-  ( A. z  z  =  y  ->  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )
4039orim1i 712 . . . 4  |-  ( ( A. z  z  =  y  \/  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )  -> 
( ( x  =  y  ->  ( [
x  /  z ]
ph  ->  [ y  / 
z ] ph )
)  \/  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) ) )
41 pm1.2 708 . . . 4  |-  ( ( ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )  \/  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )  ->  ( x  =  y  ->  ( [
x  /  z ]
ph  ->  [ y  / 
z ] ph )
) )
4240, 41syl 14 . . 3  |-  ( ( A. z  z  =  y  \/  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )  -> 
( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )
4331, 42jaoi 671 . 2  |-  ( ( A. z  z  =  x  \/  ( A. z  z  =  y  \/  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) ) )  ->  ( x  =  y  ->  ( [ x  /  z ]
ph  ->  [ y  / 
z ] ph )
) )
4421, 43ax-mp 7 1  |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    \/ wo 664   A.wal 1287   F/wnf 1394   E.wex 1426   [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693
This theorem is referenced by:  sbequ  1768
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