Theorem cbvex 1780

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 1543	. 2 |- ( ph -> A. y ph )
3		cbvex.2	. . 3 |- F/ x ps
4	3	nfri 1543	. 2 |- ( ps -> A. x ps )
5		cbvex.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	2, 4, 5	cbvexh 1779	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1484   E.wex 1516

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-nf 1485

This theorem is referenced by:  sb8e  1881  cbvex2  1947  cbvmo  2095  mo23  2096  clelab  2332  cbvrexf  2732  issetf  2781  eqvincf  2900  rexab2  2941  cbvrexcsf  3159  abn0m  3488  rabn0m  3490  euabsn  3705  eluniab  3865  cbvopab1  4122  cbvopab2  4123  cbvopab1s  4124  intexabim  4201  iinexgm  4203  opeliunxp  4735  dfdmf  4877  dfrnf  4925  elrnmpt1  4935  cbvoprab1  6027  cbvoprab2  6028  opabex3d  6216  opabex3  6217  seq3f1olemp  10673  fsum2dlemstep  11795  bdsepnfALT  15939  strcollnfALT  16036