Theorem nfeu 2064

Description: Bound-variable hypothesis builder for existential uniqueness. Note that x and y needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof rewritten by Jim Kingdon, 23-May-2018.)

Hypothesis

Ref	Expression
nfeu.1	|- F/ x ph

Assertion

Ref	Expression
nfeu	|- F/ x E! y ph

Proof of Theorem nfeu

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1542	. . 3 |- F/ z ph
2	1	sb8eu 2058	. 2 |- ( E! y ph <-> E! z [ z / y ] ph )
3		nfeu.1	. . . 4 |- F/ x ph
4	3	nfsb 1965	. . 3 |- F/ x [ z / y ] ph
5	4	nfeuv 2063	. 2 |- F/ x E! z [ z / y ] ph
6	2, 5	nfxfr 1488	1 |- F/ x E! y ph

Syntax hints:   F/wnf 1474   [wsb 1776   E!weu 2045

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048

This theorem is referenced by:  hbeu  2066  eusv2nf  4491