Theorem hbnegd 5343

Description: Deduction version of hbneg 5342.

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 1001	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
2	1	hbab 1465	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
3	2	hbneg 5342	. . 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 1284	. . . . 5 |- (ph -> A.y(y e. A -> A.x y e. A))
7		abidhb 1908	. . . . 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 5341	. . 3 |- (ph -> -u{z | A.x z e. A} = -uA)
10	9	eleq2d 1538	. 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 1102	. 2 |- (ph -> (A.x y e. -u{z | A.x z e. A} <-> A.x y e. -uA))
13	4, 10, 12	3imtr3d 541	1 |- (ph -> (y e. -uA -> A.x y e. -uA))

Syntax hints:   -> wi 3  A.wal 952   = wceq 954   e. wcel 956  {cab 1461  -ucneg 5273

This theorem is referenced by:  csbnegg 5344

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193  df-opr 3956  df-neg 5338

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