Theorem hbbrd 2709

Description: Deduction version of bound-variable hypothesis builder hbbr 2708.

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 1035	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
2	1	hbab 1503	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
3		hba1 1035	. . . . 5 |- (A.x z e. R -> A.xA.x z e. R)
4	3	hbab 1503	. . . 4 |- (y e. {z | A.x z e. R} -> A.x y e. {z | A.x z e. R})
5		hba1 1035	. . . . 5 |- (A.x z e. B -> A.xA.x z e. B)
6	5	hbab 1503	. . . 4 |- (y e. {z | A.x z e. B} -> A.x y e. {z | A.x z e. B})
7	2, 4, 6	hbbr 2708	. . 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 1319	. . . . 5 |- (ph -> A.y(y e. A -> A.x y e. A))
11		abidhb 1950	. . . . 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 1319	. . . . 5 |- (ph -> A.y(y e. B -> A.x y e. B))
15		abidhb 1950	. . . . 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 2681	. . 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 1319	. . . . 5 |- (ph -> A.y(y e. R -> A.x y e. R))
20		abidhb 1950	. . . . 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 2671	. . . 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 1136	. 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 144  A.wal 986   = wceq 988   e. wcel 990  {cab 1499   class class class wbr 2669

This theorem is referenced by:  sbcbrg 2713

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-12 1000  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-clab 1500  df-cleq 1505  df-clel 1508  df-v 1850  df-un 2094  df-sn 2457  df-pr 2458  df-op 2461  df-br 2670

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