MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbequi Structured version   Unicode version

Theorem sbequi 2089
Description: An equality theorem for substitution. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 15-Sep-2018.)
Assertion
Ref Expression
sbequi  |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )

Proof of Theorem sbequi
StepHypRef Expression
1 equtr 1745 . . 3  |-  ( z  =  x  ->  (
x  =  y  -> 
z  =  y ) )
2 sbequ2 1713 . . . 4  |-  ( z  =  x  ->  ( [ x  /  z ] ph  ->  ph ) )
3 sbequ1 1960 . . . 4  |-  ( z  =  y  ->  ( ph  ->  [ y  / 
z ] ph )
)
42, 3syl9 71 . . 3  |-  ( z  =  x  ->  (
z  =  y  -> 
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) ) )
51, 4syld 44 . 2  |-  ( z  =  x  ->  (
x  =  y  -> 
( [ x  / 
z ] ph  ->  [ y  /  z ]
ph ) ) )
6 ax13 2020 . . 3  |-  ( -.  z  =  x  -> 
( x  =  y  ->  A. z  x  =  y ) )
7 sp 1808 . . . . . 6  |-  ( A. z  z  =  x  ->  z  =  x )
87con3i 135 . . . . 5  |-  ( -.  z  =  x  ->  -.  A. z  z  =  x )
9 sb4 2070 . . . . 5  |-  ( -. 
A. z  z  =  x  ->  ( [
x  /  z ]
ph  ->  A. z ( z  =  x  ->  ph )
) )
108, 9syl 16 . . . 4  |-  ( -.  z  =  x  -> 
( [ x  / 
z ] ph  ->  A. z ( z  =  x  ->  ph ) ) )
11 equequ2 1748 . . . . . . . 8  |-  ( x  =  y  ->  (
z  =  x  <->  z  =  y ) )
1211biimprd 223 . . . . . . 7  |-  ( x  =  y  ->  (
z  =  y  -> 
z  =  x ) )
1312imim1d 75 . . . . . 6  |-  ( x  =  y  ->  (
( z  =  x  ->  ph )  ->  (
z  =  y  ->  ph ) ) )
1413al2imi 1616 . . . . 5  |-  ( A. z  x  =  y  ->  ( A. z ( z  =  x  ->  ph )  ->  A. z
( z  =  y  ->  ph ) ) )
15 sb2 2066 . . . . 5  |-  ( A. z ( z  =  y  ->  ph )  ->  [ y  /  z ] ph )
1614, 15syl6 33 . . . 4  |-  ( A. z  x  =  y  ->  ( A. z ( z  =  x  ->  ph )  ->  [ y  /  z ] ph ) )
1710, 16syl9 71 . . 3  |-  ( -.  z  =  x  -> 
( A. z  x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )
186, 17syld 44 . 2  |-  ( -.  z  =  x  -> 
( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )
195, 18pm2.61i 164 1  |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1377   [wsb 1711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-12 1803  ax-13 1968
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-nf 1600  df-sb 1712
This theorem is referenced by:  sbequ  2090
  Copyright terms: Public domain W3C validator