Theorem hbeu 2027

Description: Bound-variable hypothesis builder for uniqueness. Note that x and y needn't be distinct. (Contributed by NM, 8-Mar-1995.) (Proof rewritten by Jim Kingdon, 24-May-2018.)

Hypothesis

Ref	Expression
hbeu.1	|- ( ph -> A. x ph )

Assertion

Ref	Expression
hbeu	|- ( E! y ph -> A. x E! y ph )

Proof of Theorem hbeu

Step	Hyp	Ref	Expression
1		hbeu.1	. . . 4 |- ( ph -> A. x ph )
2	1	nfi 1442	. . 3 |- F/ x ph
3	2	nfeu 2025	. 2 |- F/ x E! y ph
4	3	nfri 1499	1 |- ( E! y ph -> A. x E! y ph )

Syntax hints:   -> wi 4   A.wal 1333   E!weu 2006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009

This theorem is referenced by:  hbmo  2045  2eu7  2100