Theorem cbvex 1767

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 1530	. 2 |- ( ph -> A. y ph )
3		cbvex.2	. . 3 |- F/ x ps
4	3	nfri 1530	. 2 |- ( ps -> A. x ps )
5		cbvex.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	2, 4, 5	cbvexh 1766	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1471   E.wex 1503

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-nf 1472

This theorem is referenced by:  sb8e  1868  cbvex2  1934  cbvmo  2082  mo23  2083  clelab  2319  cbvrexf  2719  issetf  2767  eqvincf  2886  rexab2  2927  cbvrexcsf  3145  abn0m  3473  rabn0m  3475  euabsn  3689  eluniab  3848  cbvopab1  4103  cbvopab2  4104  cbvopab1s  4105  intexabim  4182  iinexgm  4184  opeliunxp  4715  dfdmf  4856  dfrnf  4904  elrnmpt1  4914  cbvoprab1  5991  cbvoprab2  5992  opabex3d  6175  opabex3  6176  seq3f1olemp  10589  fsum2dlemstep  11580  bdsepnfALT  15451  strcollnfALT  15548