Theorem hbbrdOLD 3383

Description: Deduction version of bound-variable hypothesis builder hbbr 3381.

Hypotheses

Ref	Expression
hbbrd.1	|- (ph -> A.xph)
hbbrd.2	|- (ph -> (y e. A -> A.x y e. A))
hbbrd.3	|- (ph -> (y e. R -> A.x y e. R))
hbbrd.4	|- (ph -> (y e. B -> A.x y e. B))

Assertion

Ref	Expression
hbbrdOLD	|- (ph -> (ARB -> A.x ARB))

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

Proof of Theorem hbbrdOLD

Step	Hyp	Ref	Expression
1		hba1 1350	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
2	1	hbab 1875	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
3		hba1 1350	. . . . 5 |- (A.x z e. R -> A.xA.x z e. R)
4	3	hbab 1875	. . . 4 |- (y e. {z | A.x z e. R} -> A.x y e. {z | A.x z e. R})
5		hba1 1350	. . . . 5 |- (A.x z e. B -> A.xA.x z e. B)
6	5	hbab 1875	. . . 4 |- (y e. {z | A.x z e. B} -> A.x y e. {z | A.x z e. B})
7	2, 4, 6	hbbr 3381	. . 3 |- ({z | A.x z e. A}{z | A.x z e. R}{z | A.x z e. B} -> A.x{z | A.x z e. A}{z | A.x z e. R}{z | A.x z e. B})
8	7	a1i 8	. 2 |- (ph -> ({z | A.x z e. A}{z | A.x z e. R}{z | A.x z e. B} -> A.x{z | A.x z e. A}{z | A.x z e. R}{z | A.x z e. B}))
9		hbbrd.2	. . . . 5 |- (ph -> (y e. A -> A.x y e. A))
10	9	19.21aiv 1664	. . . 4 |- (ph -> A.y(y e. A -> A.x y e. A))
11		abidhb 2423	. . . 4 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
12	10, 11	syl 12	. . 3 |- (ph -> {z | A.x z e. A} = A)
13		hbbrd.3	. . . . 5 |- (ph -> (y e. R -> A.x y e. R))
14	13	19.21aiv 1664	. . . 4 |- (ph -> A.y(y e. R -> A.x y e. R))
15		abidhb 2423	. . . 4 |- (A.y(y e. R -> A.x y e. R) -> {z | A.x z e. R} = R)
16	14, 15	syl 12	. . 3 |- (ph -> {z | A.x z e. R} = R)
17		hbbrd.4	. . . . 5 |- (ph -> (y e. B -> A.x y e. B))
18	17	19.21aiv 1664	. . . 4 |- (ph -> A.y(y e. B -> A.x y e. B))
19		abidhb 2423	. . . 4 |- (A.y(y e. B -> A.x y e. B) -> {z | A.x z e. B} = B)
20	18, 19	syl 12	. . 3 |- (ph -> {z | A.x z e. B} = B)
21	12, 16, 20	breq123d 3353	. 2 |- (ph -> ({z | A.x z e. A}{z | A.x z e. R}{z | A.x z e. B} <-> ARB))
22		hbbrd.1	. . 3 |- (ph -> A.xph)
23	22, 21	albid 1459	. 2 |- (ph -> (A.x{z | A.x z e. A}{z | A.x z e. R}{z | A.x z e. B} <-> A.x ARB))
24	8, 21, 23	3imtr3d 601	1 |- (ph -> (ARB -> A.x ARB))

Syntax hints:   -> wi 3  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871   class class class wbr 3338

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339

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