Theorem hboprd 3988

Description: Deduction version of bound-variable hypothesis builder hbopr 3987.

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 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. F -> A.xA.x z e. F)
4	3	hbab 1470	. . . 4 |- (y e. {z | A.x z e. F} -> A.x y e. {z | A.x z e. F})
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	hbopr 3987	. . 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 1288	. . . . . 6 |- (ph -> A.y(y e. A -> A.x y e. A))
11		abidhb 1915	. . . . . 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 1288	. . . . . 6 |- (ph -> A.y(y e. B -> A.x y e. B))
15		abidhb 1915	. . . . . 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 3984	. . . 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 1288	. . . . 5 |- (ph -> A.y(y e. F -> A.x y e. F))
20		abidhb 1915	. . . . 5 |- (A.y(y e. F -> A.x y e. F) -> {z | A.x z e. F} = F)
21		opreq 3973	. . . . 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 1510	. . 3 |- (ph -> ({z | A.x z e. A}{z | A.x z e. F}{z | A.x z e. B}) = (AFB))
24	23	eleq2d 1544	. 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 1106	. 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 544	1 |- (ph -> (y e. (AFB) -> A.x y e. (AFB)))

Syntax hints:   -> wi 3  A.wal 956   = wceq 958   e. wcel 960  {cab 1466  (class class class)co 3969

This theorem is referenced by:  csboprg 3992

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-11 969  ax-12 970  ax-13 971  ax-14 972  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  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971

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