MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbco3 Structured version   Unicode version

Theorem sbco3 2163
Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbco3  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )

Proof of Theorem sbco3
StepHypRef Expression
1 drsb1 2102 . . 3  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] [ y  /  x ] ph  <->  [ z  /  y ] [
y  /  x ] ph ) )
2 sbequ12a 1946 . . . . 5  |-  ( x  =  y  ->  ( [ y  /  x ] ph  <->  [ x  /  y ] ph ) )
32alimi 1568 . . . 4  |-  ( A. x  x  =  y  ->  A. x ( [ y  /  x ] ph 
<->  [ x  /  y ] ph ) )
4 spsbbi 2151 . . . 4  |-  ( A. x ( [ y  /  x ] ph  <->  [ x  /  y ]
ph )  ->  ( [ z  /  x ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph ) )
53, 4syl 16 . . 3  |-  ( A. x  x  =  y  ->  ( [ z  /  x ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph ) )
61, 5bitr3d 247 . 2  |-  ( A. x  x  =  y  ->  ( [ z  / 
y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph ) )
7 sbco 2158 . . . 4  |-  ( [ x  /  y ] [ y  /  x ] ph  <->  [ x  /  y ] ph )
87sbbii 1665 . . 3  |-  ( [ z  /  x ] [ x  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
9 nfnae 2044 . . . 4  |-  F/ y  -.  A. x  x  =  y
10 nfnae 2044 . . . 4  |-  F/ x  -.  A. x  x  =  y
11 nfsb2 2092 . . . 4  |-  ( -. 
A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
129, 10, 11sbco2d 2162 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
z  /  x ] [ x  /  y ] [ y  /  x ] ph  <->  [ z  /  y ] [ y  /  x ] ph ) )
138, 12syl5rbbr 252 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph ) )
146, 13pm2.61i 158 1  |-  ( [ z  /  y ] [ y  /  x ] ph  <->  [ z  /  x ] [ x  /  y ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177   A.wal 1549   [wsb 1658
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-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
  Copyright terms: Public domain W3C validator