Theorem hbeud 2022

Description: Deduction version of hbeu 2021. (Contributed by NM, 15-Feb-2013.) (Proof rewritten by Jim Kingdon, 25-May-2018.)

Hypotheses

Ref	Expression
hbeud.1	|- ( ph -> A. x ph )
hbeud.2	|- ( ph -> A. y ph )
hbeud.3	|- ( ph -> ( ps -> A. x ps ) )

Assertion

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

Proof of Theorem hbeud

Step	Hyp	Ref	Expression
1		hbeud.2	. . . 4 |- ( ph -> A. y ph )
2	1	nfi 1439	. . 3 |- F/ y ph
3		hbeud.1	. . . . 5 |- ( ph -> A. x ph )
4	3	nfi 1439	. . . 4 |- F/ x ph
5		hbeud.3	. . . 4 |- ( ph -> ( ps -> A. x ps ) )
6	4, 5	nfd 1504	. . 3 |- ( ph -> F/ x ps )
7	2, 6	nfeud 2016	. 2 |- ( ph -> F/ x E! y ps )
8	7	nfrd 1501	1 |- ( ph -> ( E! y ps -> A. x E! y ps ) )

Syntax hints:   -> wi 4   A.wal 1330   E!weu 2000

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003

This theorem is referenced by: (None)