Theorem cbvex 1770

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 1533	. 2 |- ( ph -> A. y ph )
3		cbvex.2	. . 3 |- F/ x ps
4	3	nfri 1533	. 2 |- ( ps -> A. x ps )
5		cbvex.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	2, 4, 5	cbvexh 1769	1 |- ( E. x ph <-> E. y ps )

Syntax hints:   -> wi 4   <-> wb 105   F/wnf 1474   E.wex 1506

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  sb8e  1871  cbvex2  1937  cbvmo  2085  mo23  2086  clelab  2322  cbvrexf  2722  issetf  2770  eqvincf  2889  rexab2  2930  cbvrexcsf  3148  abn0m  3476  rabn0m  3478  euabsn  3692  eluniab  3851  cbvopab1  4106  cbvopab2  4107  cbvopab1s  4108  intexabim  4185  iinexgm  4187  opeliunxp  4718  dfdmf  4859  dfrnf  4907  elrnmpt1  4917  cbvoprab1  5994  cbvoprab2  5995  opabex3d  6178  opabex3  6179  seq3f1olemp  10607  fsum2dlemstep  11599  bdsepnfALT  15535  strcollnfALT  15632