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

Theorem cbvab 2551
Description: Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
Hypotheses
Ref Expression
cbvab.1  |-  F/ y
ph
cbvab.2  |-  F/ x ps
cbvab.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvab  |-  { x  |  ph }  =  {
y  |  ps }

Proof of Theorem cbvab
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cbvab.2 . . . . 5  |-  F/ x ps
21nfsb 2148 . . . 4  |-  F/ x [ z  /  y ] ps
3 cbvab.1 . . . . . 6  |-  F/ y
ph
4 cbvab.3 . . . . . . . 8  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
54equcoms 1732 . . . . . . 7  |-  ( y  =  x  ->  ( ph 
<->  ps ) )
65bicomd 201 . . . . . 6  |-  ( y  =  x  ->  ( ps 
<-> 
ph ) )
73, 6sbie 2098 . . . . 5  |-  ( [ x  /  y ] ps  <->  ph )
8 sbequ 2064 . . . . 5  |-  ( x  =  z  ->  ( [ x  /  y ] ps  <->  [ z  /  y ] ps ) )
97, 8syl5bbr 259 . . . 4  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  y ] ps ) )
102, 9sbie 2098 . . 3  |-  ( [ z  /  x ] ph 
<->  [ z  /  y ] ps )
11 df-clab 2420 . . 3  |-  ( z  e.  { x  | 
ph }  <->  [ z  /  x ] ph )
12 df-clab 2420 . . 3  |-  ( z  e.  { y  |  ps }  <->  [ z  /  y ] ps )
1310, 11, 123bitr4i 277 . 2  |-  ( z  e.  { x  | 
ph }  <->  z  e.  { y  |  ps }
)
1413eqriv 2430 1  |-  { x  |  ph }  =  {
y  |  ps }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1362   F/wnf 1592   [wsb 1699    e. wcel 1755   {cab 2419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426
This theorem is referenced by:  cbvabv  2552  cbvrab  2960  cbvsbc  3203  cbvrabcsf  3310  rabsnifsb  3931  dfdmf  5020  dfrnf  5065  funfv2f  5748  abrexex2g  6543  abrexex2  6547  ptrest  28266  rabasiun  30073  bnj873  31616
  Copyright terms: Public domain W3C validator