Theorem hbeu 1382

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 1136	. . . . . 6 |- (A.y y = x -> (A.yA.y(ph <-> y = z) -> A.xA.y(ph <-> y = z)))
2	1	alequcoms 1139	. . . . 5 |- (A.x x = y -> (A.yA.y(ph <-> y = z) -> A.xA.y(ph <-> y = z)))
3		hba1 1000	. . . . 5 |- (A.y(ph <-> y = z) -> A.yA.y(ph <-> y = z))
4	2, 3	syl5 21	. . . 4 |- (A.x x = y -> (A.y(ph <-> y = z) -> A.xA.y(ph <-> y = z)))
5		hbnae 1143	. . . . 5 |- (-. A.x x = y -> A.y -. A.x x = y)
6		hbnae 1143	. . . . . 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 1348	. . . . . 6 |- (-. A.x x = y -> (y = z -> A.x y = z))
10	6, 8, 9	hbbid 1108	. . . . 5 |- (-. A.x x = y -> ((ph <-> y = z) -> A.x(ph <-> y = z)))
11	5, 10	hbald 1109	. . . 4 |- (-. A.x x = y -> (A.y(ph <-> y = z) -> A.xA.y(ph <-> y = z)))
12	4, 11	pm2.61i 126	. . 3 |- (A.y(ph <-> y = z) -> A.xA.y(ph <-> y = z))
13	12	hbex 1003	. 2 |- (E.zA.y(ph <-> y = z) -> A.xE.zA.y(ph <-> y = z))
14		df-eu 1375	. 2 |- (E!yph <-> E.zA.y(ph <-> y = z))
15	14	albii 996	. 2 |- (A.xE!yph <-> A.xE.zA.y(ph <-> y = z))
16	13, 14, 15	3imtr4 219	1 |- (E!yph -> A.xE!yph)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146  A.wal 951  E.wex 977  E!weu 1373

This theorem is referenced by:  hbmo 1400  2eu7 1448  2eu8 1449

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-eu 1375

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