Theorem cbvexv 1928

Description: Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
cbvalv.1	|- ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cbvexv	|- ( E. x ph <-> E. y ps )

Distinct variable groups:   ph, y   ps, x

Allowed substitution hints:   ph( x)   ps( y)

Proof of Theorem cbvexv

Step	Hyp	Ref	Expression
1		ax-17 1536	. 2 |- ( ph -> A. y ph )
2		ax-17 1536	. 2 |- ( ps -> A. x ps )
3		cbvalv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvexh 1765	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1502

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-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  eujust  2038  euind  2936  reuind  2954  r19.2m  3521  r19.3rm  3523  r19.9rmv  3526  raaanlem  3540  raaan  3541  cbvopab2v  4092  bm1.3ii  4136  mss  4238  zfun  4446  xpiindim  4776  relop  4789  dmmrnm  4858  dmxpm  4859  dmcoss  4908  xpm  5062  cnviinm  5182  iotam  5220  fv3  5550  fo1stresm  6175  fo2ndresm  6176  tfr1onlemaccex  6362  tfrcllemaccex  6375  iinerm  6620  riinerm  6621  ixpiinm  6737  ac6sfi  6911  ctmlemr  7120  ctm  7121  ctssdclemr  7124  ctssdc  7125  fodjum  7157  acfun  7219  ccfunen  7276  cc2lem  7278  cc2  7279  ltexprlemdisj  7618  ltexprlemloc  7619  recexprlemdisj  7642  suplocsr  7821  axpre-suploc  7914  zfz1isolem1  10833  climmo  11319  summodc  11404  nninfdcex  11967  zsupssdc  11968  ctiunct  12454  ismnd  12841  dfgrp3me  12996  issubg2m  13080  subrgintm  13458  islssm  13541  islidlm  13663  neipsm  13925  suplociccex  14374  bdbm1.3ii  14914