Theorem cbvex 1809

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 1572	. 2 |- ( ph -> A. y ph )
3		cbvex.2	. . 3 |- F/ x ps
4	3	nfri 1572	. 2 |- ( ps -> A. x ps )
5		cbvex.3	. 2 |- ( x = y -> ( ph <-> ps ) )
6	2, 4, 5	cbvexh 1808	1 |- ( E. x ph <-> E. y ps )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   F/wnf 1513   E.wex 1545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This proof depends on definitions:  df-bi 117  df-nf 1514

This theorem is used by:  sb8e  1910  cbvex2  1978  cbvmo  2126  mo23  2128  clelab  2366  cbvrexf  2778  issetf  2829  eqvincf  2951  rexab2  2992  cbvrexcsf  3211  abn0m  3547  rabn0m  3549  euabsn  3781  eluniab  3947  cbvopab1  4204  cbvopab2  4205  cbvopab1s  4206  intexabim  4288  iinexgm  4290  opeliunxp  4830  dfdmf  4974  dfrnf  5023  elrnmpt1  5033  cbvoprab1  6160  cbvoprab2  6161  opabex3d  6350  opabex3  6351  seq3f1olemp  10952  fsum2dlemstep  12201  bdsepnfALT  16915  strcollnfALT  17012