Theorem hbeu 1782

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-10o 1500	. . . . . 6 |- (A.y y = x -> (A.yA.y(ph <-> y = z) -> A.xA.y(ph <-> y = z)))
2	1	alequcoms 1503	. . . . 5 |- (A.x x = y -> (A.yA.y(ph <-> y = z) -> A.xA.y(ph <-> y = z)))
3		hba1 1350	. . . . 5 |- (A.y(ph <-> y = z) -> A.yA.y(ph <-> y = z))
4	2, 3	syl5 20	. . . 4 |- (A.x x = y -> (A.y(ph <-> y = z) -> A.xA.y(ph <-> y = z)))
5		hbnae 1507	. . . . 5 |- (-. A.x x = y -> A.y -. A.x x = y)
6		hbnae 1507	. . . . . 6 |- (-. A.x x = y -> A.x -. A.x x = y)
7		hbeu.1	. . . . . . 7 |- (ph -> A.xph)
8	7	a1i 8	. . . . . 6 |- (-. A.x x = y -> (ph -> A.xph))
9		dveeq1 1745	. . . . . 6 |- (-. A.x x = y -> (y = z -> A.x y = z))
10	6, 8, 9	hbbid 1470	. . . . 5 |- (-. A.x x = y -> ((ph <-> y = z) -> A.x(ph <-> y = z)))
11	5, 10	hbald 1471	. . . 4 |- (-. A.x x = y -> (A.y(ph <-> y = z) -> A.xA.y(ph <-> y = z)))
12	4, 11	pm2.61i 140	. . 3 |- (A.y(ph <-> y = z) -> A.xA.y(ph <-> y = z))
13	12	hbex 1353	. 2 |- (E.zA.y(ph <-> y = z) -> A.xE.zA.y(ph <-> y = z))
14		df-eu 1775	. 2 |- (E!yph <-> E.zA.y(ph <-> y = z))
15	14	albii 1346	. 2 |- (A.xE!yph <-> A.xE.zA.y(ph <-> y = z))
16	13, 14, 15	3imtr4i 236	1 |- (E!yph -> A.xE!yph)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163  A.wal 1296  E.wex 1326  E!weu 1771

This theorem is referenced by:  hbmo 1803  2eu7 1859  2eu8 1860  hbreu 2251  eualexeq 3825  euexeqOLD 3826  euexaleq 3827  eufromeq1 3828  bnj1332 13499

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-eu 1775

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