Theorem hbbrd 2664

Description: Deduction version of bound-variable hypothesis builder hbbr 2663.

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 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		hba1 1005	. . . . 5 |- (A.x z e. R -> A.xA.x z e. R)
4	3	hbab 1470	. . . 4 |- (y e. {z | A.x z e. R} -> A.x y e. {z | A.x z e. R})
5		hba1 1005	. . . . 5 |- (A.x z e. B -> A.xA.x z e. B)
6	5	hbab 1470	. . . 4 |- (y e. {z | A.x z e. B} -> A.x y e. {z | A.x z e. B})
7	2, 4, 6	hbbr 2663	. . 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 1288	. . . . 5 |- (ph -> A.y(y e. A -> A.x y e. A))
11		abidhb 1915	. . . . 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 1288	. . . . 5 |- (ph -> A.y(y e. B -> A.x y e. B))
15		abidhb 1915	. . . . 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 2636	. . 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 1288	. . . . 5 |- (ph -> A.y(y e. R -> A.x y e. R))
20		abidhb 1915	. . . . 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 2626	. . . 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 530	. 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 1106	. 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 544	1 |- (ph -> (ARB -> A.x ARB))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 956   = wceq 958   e. wcel 960  {cab 1466   class class class wbr 2624

This theorem is referenced by:  sbcbrg 2667

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-12 970  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625

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