Theorem cbvexv 1836

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 1459	. 2 |- ( ph -> A. y ph )
2		ax-17 1459	. 2 |- ( ps -> A. x ps )
3		cbvalv.1	. 2 |- ( x = y -> ( ph <-> ps ) )
4	1, 2, 3	cbvexh 1678	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -> wi 4   <-> wb 103   E.wex 1421

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  eujust  1943  euind  2779  reuind  2795  r19.3rm  3330  r19.9rmv  3333  raaanlem  3346  raaan  3347  cbvopab2v  3855  bm1.3ii  3899  mss  3981  zfun  4189  xpiindim  4491  relop  4504  dmmrnm  4572  dmxpm  4573  dmcoss  4619  xpm  4765  cnviinm  4879  fv3  5218  fo1stresm  5808  fo2ndresm  5809  iinerm  6201  riinerm  6202  ac6sfi  6379  ltexprlemdisj  6796  ltexprlemloc  6797  recexprlemdisj  6820  climmo  10137  bdbm1.3ii  10682