Theorem nfeu 1919

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 1421	. . 3 |- F/ z ph
2	1	sb8eu 1913	. 2 |- ( E! y ph <-> E! z [ z / y ] ph )
3		nfeu.1	. . . 4 |- F/ x ph
4	3	nfsb 1822	. . 3 |- F/ x [ z / y ] ph
5	4	nfeuv 1918	. 2 |- F/ x E! z [ z / y ] ph
6	2, 5	nfxfr 1363	1 |- F/ x E! y ph

Syntax hints:   F/wnf 1349   [wsb 1645   E!weu 1900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903

This theorem is referenced by:  hbeu  1921  eusv2nf  4188