Theorem hbeu 2100

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 1511	. . 3 |- F/ x ph
3	2	nfeu 2098	. 2 |- F/ x E! y ph
4	3	nfri 1568	1 |- ( E! y ph -> A. x E! y ph )

Syntax hints:   -> wi 4   A.wal 1396   E!weu 2079

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082

This theorem is referenced by:  hbmo  2118  2eu7  2174