Theorem hbeu 1016

Description: Bound-variable hypothesis builder for "at most one". Note that x and y needn't be distinct (this makes the proof more difficult).

Hypothesis

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

Assertion

Ref	Expression
hbeu	|- (E!yph -> A.xE!yph)

Proof of Theorem hbeu

Step	Hyp	Ref	Expression
1		ax-10 800	. . . . . 6 |- (A.y y = x -> (A.yA.y(ph <-> y = z) -> A.xA.y(ph <-> y = z)))
2	1	eq4s 822	. . . . 5 |- (A.x x = y -> (A.yA.y(ph <-> y = z) -> A.xA.y(ph <-> y = z)))
3		hba1 698	. . . . 5 |- (A.y(ph <-> y = z) -> A.yA.y(ph <-> y = z))
4	2, 3	syl5 22	. . . 4 |- (A.x x = y -> (A.y(ph <-> y = z) -> A.xA.y(ph <-> y = z)))
5		eq6 826	. . . . 5 |- (-. A.x x = y -> A.y -. A.x x = y)
6		eq6 826	. . . . . 6 |- (-. A.x x = y -> A.x -. A.x x = y)
7		hbeu.1	. . . . . . 7 |- (ph -> A.xph)
8	7	a1i 7	. . . . . 6 |- (-. A.x x = y -> (ph -> A.xph))
9		ddeeq1 1001	. . . . . 6 |- (-. A.x x = y -> (y = z -> A.x y = z))
10	6, 8, 9	hbbid 789	. . . . 5 |- (-. A.x x = y -> ((ph <-> y = z) -> A.x(ph <-> y = z)))
11	5, 10	hbald 790	. . . 4 |- (-. A.x x = y -> (A.y(ph <-> y = z) -> A.xA.y(ph <-> y = z)))
12	4, 11	pm2.61i 110	. . 3 |- (A.y(ph <-> y = z) -> A.xA.y(ph <-> y = z))
13	12	hbex 701	. 2 |- (E.zA.y(ph <-> y = z) -> A.xE.zA.y(ph <-> y = z))
14		df-eu 1009	. 2 |- (E!yph <-> E.zA.y(ph <-> y = z))
15	14	bial 695	. 2 |- (A.xE!yph <-> A.xE.zA.y(ph <-> y = z))
16	13, 14, 15	3imtr4 192	1 |- (E!yph -> A.xE!yph)

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127  A.wal 672  E.wex 678   = weq 797  E!weu 1007

This theorem is referenced by:  hbmo 1033

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-eu 1009

metamath.org