Theorem cbvex 1802

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 1565	. 2 |- ( ph -> A. y ph )
3		cbvex.2	. . 3 |- F/ x ps
4	3	nfri 1565	. 2 |- ( ps -> A. x ps )
5		cbvex.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	2, 4, 5	cbvexh 1801	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1506   E.wex 1538

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507

This theorem is referenced by:  sb8e  1903  cbvex2  1969  cbvmo  2117  mo23  2119  clelab  2355  cbvrexf  2757  issetf  2807  eqvincf  2928  rexab2  2969  cbvrexcsf  3188  abn0m  3517  rabn0m  3519  euabsn  3736  eluniab  3899  cbvopab1  4156  cbvopab2  4157  cbvopab1s  4158  intexabim  4235  iinexgm  4237  opeliunxp  4771  dfdmf  4913  dfrnf  4961  elrnmpt1  4971  cbvoprab1  6067  cbvoprab2  6068  opabex3d  6256  opabex3  6257  seq3f1olemp  10724  fsum2dlemstep  11931  bdsepnfALT  16182  strcollnfALT  16279