Theorem hbnegd 5375

Description: Deduction version of hbneg 5374.

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		hba1 1005	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
2	1	hbab 1470	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
3	2	hbneg 5374	. . 3 |- (y e. -u{z | A.x z e. A} -> A.x y e. -u{z | A.x z e. A})
4	3	a1i 8	. 2 |- (ph -> (y e. -u{z | A.x z e. A} -> A.x y e. -u{z | A.x z e. A}))
5		hbnegd.2	. . . . . 6 |- (ph -> (y e. A -> A.x y e. A))
6	5	19.21aiv 1288	. . . . 5 |- (ph -> A.y(y e. A -> A.x y e. A))
7		abidhb 1915	. . . . 5 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
8	6, 7	syl 10	. . . 4 |- (ph -> {z | A.x z e. A} = A)
9	8	negeqd 5373	. . 3 |- (ph -> -u{z | A.x z e. A} = -uA)
10	9	eleq2d 1544	. 2 |- (ph -> (y e. -u{z | A.x z e. A} <-> y e. -uA))
11		hbnegd.1	. . 3 |- (ph -> A.xph)
12	11, 10	albid 1106	. 2 |- (ph -> (A.x y e. -u{z | A.x z e. A} <-> A.x y e. -uA))
13	4, 10, 12	3imtr3d 544	1 |- (ph -> (y e. -uA -> A.x y e. -uA))

Syntax hints:   -> wi 3  A.wal 956   = wceq 958   e. wcel 960  {cab 1466  -ucneg 5305

This theorem is referenced by:  csbnegg 5376

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971  df-neg 5370

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