Theorem hbbrd 2659

Description: Deduction version of bound-variable hypothesis builder hbbr 2658.

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
hbbrd	|- (ph -> (ARB -> A.x ARB))

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

Proof of Theorem hbbrd

Step	Hyp	Ref	Expression
1		hba1 1003	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
2	1	hbab 1467	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
3		hba1 1003	. . . . 5 |- (A.x z e. R -> A.xA.x z e. R)
4	3	hbab 1467	. . . 4 |- (y e. {z | A.x z e. R} -> A.x y e. {z | A.x z e. R})
5		hba1 1003	. . . . 5 |- (A.x z e. B -> A.xA.x z e. B)
6	5	hbab 1467	. . . 4 |- (y e. {z | A.x z e. B} -> A.x y e. {z | A.x z e. B})
7	2, 4, 6	hbbr 2658	. . 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	. . . . . 6 |- (ph -> (y e. A -> A.x y e. A))
10	9	19.21aiv 1286	. . . . 5 |- (ph -> A.y(y e. A -> A.x y e. A))
11		abidhb 1912	. . . . 5 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
12	10, 11	syl 10	. . . 4 |- (ph -> {z | A.x z e. A} = A)
13		hbbrd.4	. . . . . 6 |- (ph -> (y e. B -> A.x y e. B))
14	13	19.21aiv 1286	. . . . 5 |- (ph -> A.y(y e. B -> A.x y e. B))
15		abidhb 1912	. . . . 5 |- (A.y(y e. B -> A.x y e. B) -> {z | A.x z e. B} = B)
16	14, 15	syl 10	. . . 4 |- (ph -> {z | A.x z e. B} = B)
17	12, 16	breq12d 2631	. . 3 |- (ph -> ({z | A.x z e. A}{z | A.x z e. R}{z | A.x z e. B} <-> A{z | A.x z e. R}B))
18		hbbrd.3	. . . . . 6 |- (ph -> (y e. R -> A.x y e. R))
19	18	19.21aiv 1286	. . . . 5 |- (ph -> A.y(y e. R -> A.x y e. R))
20		abidhb 1912	. . . . 5 |- (A.y(y e. R -> A.x y e. R) -> {z | A.x z e. R} = R)
21	19, 20	syl 10	. . . 4 |- (ph -> {z | A.x z e. R} = R)
22		breq 2621	. . . 4 |- ({z | A.x z e. R} = R -> (A{z | A.x z e. R}B <-> ARB))
23	21, 22	syl 10	. . 3 |- (ph -> (A{z | A.x z e. R}B <-> ARB))
24	17, 23	bitrd 528	. 2 |- (ph -> ({z | A.x z e. A}{z | A.x z e. R}{z | A.x z e. B} <-> ARB))
25		hbbrd.1	. . 3 |- (ph -> A.xph)
26	25, 24	albid 1104	. 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))
27	8, 24, 26	3imtr3d 542	1 |- (ph -> (ARB -> A.x ARB))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  {cab 1463   class class class wbr 2619

This theorem is referenced by:  sbcbrg 2662

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620

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