Theorem hboprd 3973

Description: Deduction version of bound-variable hypothesis builder hbopr 3972.

Hypotheses

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

Assertion

Ref	Expression
hboprd	|- (ph -> (y e. (AFB) -> A.x y e. (AFB)))

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

Proof of Theorem hboprd

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		hba1 1001	. . . . 5 |- (A.x z e. F -> A.xA.x z e. F)
4	3	hbab 1465	. . . 4 |- (y e. {z | A.x z e. F} -> A.x y e. {z | A.x z e. F})
5		hba1 1001	. . . . 5 |- (A.x z e. B -> A.xA.x z e. B)
6	5	hbab 1465	. . . 4 |- (y e. {z | A.x z e. B} -> A.x y e. {z | A.x z e. B})
7	2, 4, 6	hbopr 3972	. . 3 |- (y e. ({z | A.x z e. A}{z | A.x z e. F}{z | A.x z e. B}) -> A.x y e. ({z | A.x z e. A}{z | A.x z e. F}{z | A.x z e. B}))
8	7	a1i 8	. 2 |- (ph -> (y e. ({z | A.x z e. A}{z | A.x z e. F}{z | A.x z e. B}) -> A.x y e. ({z | A.x z e. A}{z | A.x z e. F}{z | A.x z e. B})))
9		hboprd.2	. . . . . . 7 |- (ph -> (y e. A -> A.x y e. A))
10	9	19.21aiv 1284	. . . . . 6 |- (ph -> A.y(y e. A -> A.x y e. A))
11		abidhb 1908	. . . . . 6 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
12	10, 11	syl 10	. . . . 5 |- (ph -> {z | A.x z e. A} = A)
13		hboprd.4	. . . . . . 7 |- (ph -> (y e. B -> A.x y e. B))
14	13	19.21aiv 1284	. . . . . 6 |- (ph -> A.y(y e. B -> A.x y e. B))
15		abidhb 1908	. . . . . 6 |- (A.y(y e. B -> A.x y e. B) -> {z | A.x z e. B} = B)
16	14, 15	syl 10	. . . . 5 |- (ph -> {z | A.x z e. B} = B)
17	12, 16	opreq12d 3969	. . . 4 |- (ph -> ({z | A.x z e. A}{z | A.x z e. F}{z | A.x z e. B}) = (A{z | A.x z e. F}B))
18		hboprd.3	. . . . . 6 |- (ph -> (y e. F -> A.x y e. F))
19	18	19.21aiv 1284	. . . . 5 |- (ph -> A.y(y e. F -> A.x y e. F))
20		abidhb 1908	. . . . 5 |- (A.y(y e. F -> A.x y e. F) -> {z | A.x z e. F} = F)
21		opreq 3958	. . . . 5 |- ({z | A.x z e. F} = F -> (A{z | A.x z e. F}B) = (AFB))
22	19, 20, 21	3syl 20	. . . 4 |- (ph -> (A{z | A.x z e. F}B) = (AFB))
23	17, 22	eqtrd 1504	. . 3 |- (ph -> ({z | A.x z e. A}{z | A.x z e. F}{z | A.x z e. B}) = (AFB))
24	23	eleq2d 1538	. 2 |- (ph -> (y e. ({z | A.x z e. A}{z | A.x z e. F}{z | A.x z e. B}) <-> y e. (AFB)))
25		hboprd.1	. . 3 |- (ph -> A.xph)
26	25, 24	albid 1102	. 2 |- (ph -> (A.x y e. ({z | A.x z e. A}{z | A.x z e. F}{z | A.x z e. B}) <-> A.x y e. (AFB)))
27	8, 24, 26	3imtr3d 541	1 |- (ph -> (y e. (AFB) -> A.x y e. (AFB)))

Syntax hints:   -> wi 3  A.wal 952   = wceq 954   e. wcel 956  {cab 1461  (class class class)co 3954

This theorem is referenced by:  csboprg 3977

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

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