Theorem cbvex 1804

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

Hypotheses

Ref	Expression
cbvex.1	|- F/ y ph
cbvex.2	|- F/ x ps
cbvex.3	|- ( x = y -> ( ph <-> ps ) )

Assertion

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

Proof of Theorem cbvex

Step	Hyp	Ref	Expression
1		cbvex.1	. . 3 |- F/ y ph
2	1	nfri 1567	. 2 |- ( ph -> A. y ph )
3		cbvex.2	. . 3 |- F/ x ps
4	3	nfri 1567	. 2 |- ( ps -> A. x ps )
5		cbvex.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	2, 4, 5	cbvexh 1803	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1508   E.wex 1540

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-nf 1509

This theorem is referenced by:  sb8e  1905  cbvex2  1971  cbvmo  2119  mo23  2121  clelab  2357  cbvrexf  2759  issetf  2810  eqvincf  2931  rexab2  2972  cbvrexcsf  3191  abn0m  3520  rabn0m  3522  euabsn  3741  eluniab  3905  cbvopab1  4162  cbvopab2  4163  cbvopab1s  4164  intexabim  4242  iinexgm  4244  opeliunxp  4781  dfdmf  4924  dfrnf  4973  elrnmpt1  4983  cbvoprab1  6092  cbvoprab2  6093  opabex3d  6282  opabex3  6283  seq3f1olemp  10776  fsum2dlemstep  11994  bdsepnfALT  16484  strcollnfALT  16581