ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cbvral Unicode version

Theorem cbvral 2699
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.)
Hypotheses
Ref Expression
cbvral.1  |-  F/ y
ph
cbvral.2  |-  F/ x ps
cbvral.3  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvral  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )
Distinct variable groups:    x, A    y, A
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem cbvral
StepHypRef Expression
1 nfcv 2319 . 2  |-  F/_ x A
2 nfcv 2319 . 2  |-  F/_ y A
3 cbvral.1 . 2  |-  F/ y
ph
4 cbvral.2 . 2  |-  F/ x ps
5 cbvral.3 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
61, 2, 3, 4, 5cbvralf 2696 1  |-  ( A. x  e.  A  ph  <->  A. y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105   F/wnf 1460   A.wral 2455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460
This theorem is referenced by:  cbvralv  2703  cbvralsv  2719  cbviin  3922  frind  4348  ralxpf  4768  eqfnfv2f  5612  ralrnmpt  5653  dff13f  5764  ofrfval2  6092  fmpox  6194  cbvixp  6708  mptelixpg  6727  xpf1o  6837  indstr  9569  fsum3  11366  fsum00  11441  mertenslem2  11515  fprodcl2lem  11584  fprodle  11619  ctiunctal  12412  cnmpt11  13416  cnmpt21  13424  bj-bdfindes  14323  bj-findes  14355
  Copyright terms: Public domain W3C validator