Theorem hbnegd 6518

Description: Deduction version of hbneg 6517. (The proof was shortened by Andrew Salmon, 22-Oct-2011.)

Hypotheses

Ref	Expression
hbnegd.1	|- (ph -> A.xph)
hbnegd.2	|- (ph -> (y e. A -> A.x y e. A))

Assertion

Ref	Expression
hbnegd	|- (ph -> (y e. -uA -> A.x y e. -uA))

Distinct variable groups:   y,A   ph,y   x,y

Proof of Theorem hbnegd

Step	Hyp	Ref	Expression
1		hbnegd.1	. . 3 |- (ph -> A.xph)
2		ax-17 1317	. . . 4 |- (y e. 0 -> A.x y e. 0)
3	2	a1i 8	. . 3 |- (ph -> (y e. 0 -> A.x y e. 0))
4		ax-17 1317	. . . 4 |- (y e. - -> A.x y e. - )
5	4	a1i 8	. . 3 |- (ph -> (y e. - -> A.x y e. - ))
6		hbnegd.2	. . 3 |- (ph -> (y e. A -> A.x y e. A))
7	1, 3, 5, 6	hboprd 4905	. 2 |- (ph -> (y e. (0 - A) -> A.x y e. (0 - A)))
8		df-neg 6513	. . 3 |- -uA = (0 - A)
9	8	eleq2i 1961	. 2 |- (y e. -uA <-> y e. (0 - A))
10	9	albii 1346	. 2 |- (A.x y e. -uA <-> A.x y e. (0 - A))
11	7, 9, 10	3imtr4g 612	1 |- (ph -> (y e. -uA -> A.x y e. -uA))

Syntax hints:   -> wi 3  A.wal 1296   e. wcel 1300  (class class class)co 4884  0cc0 6386   - cmin 6445  -ucneg 6446

This theorem is referenced by:  csbneggOLD 6521

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-neg 6513

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