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Theorem cbvralsv 2952
Description: Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
cbvralsv  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  [ y  /  x ] ph )
Distinct variable groups:    x, A    y, A    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem cbvralsv
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1631 . . 3  |-  F/ z
ph
2 nfs1v 2189 . . 3  |-  F/ x [ z  /  x ] ph
3 sbequ12 1948 . . 3  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
41, 2, 3cbvral 2937 . 2  |-  ( A. x  e.  A  ph  <->  A. z  e.  A  [ z  /  x ] ph )
5 nfv 1631 . . . 4  |-  F/ y
ph
65nfsb 2192 . . 3  |-  F/ y [ z  /  x ] ph
7 nfv 1631 . . 3  |-  F/ z [ y  /  x ] ph
8 sbequ 2116 . . 3  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
96, 7, 8cbvral 2937 . 2  |-  ( A. z  e.  A  [
z  /  x ] ph 
<-> 
A. y  e.  A  [ y  /  x ] ph )
104, 9bitri 242 1  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   [wsb 1660   A.wral 2712
This theorem is referenced by:  sbralie  2954  rspsbc  3258  tfinds  4874  tfindes  4877  ralxpf  5054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ral 2717
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