Theorem hbopd 2501

Description: Deduction version of bound-variable hypothesis builder hbop 2500.

Hypotheses

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

Assertion

Ref	Expression
hbopd	|- (ph -> (y e. <.A, B>. -> A.x y e. <.A, B>.))

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

Proof of Theorem hbopd

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. B -> A.xA.x z e. B)
4	3	hbab 1470	. . . 4 |- (y e. {z | A.x z e. B} -> A.x y e. {z | A.x z e. B})
5	2, 4	hbop 2500	. . 3 |- (y e. <.{z | A.x z e. A}, {z | A.x z e. B}>. -> A.x y e. <.{z | A.x z e. A}, {z | A.x z e. B}>.)
6	5	a1i 8	. 2 |- (ph -> (y e. <.{z | A.x z e. A}, {z | A.x z e. B}>. -> A.x y e. <.{z | A.x z e. A}, {z | A.x z e. B}>.))
7		hbopd.2	. . . . . . 7 |- (ph -> (y e. A -> A.x y e. A))
8	7	19.21aiv 1288	. . . . . 6 |- (ph -> A.y(y e. A -> A.x y e. A))
9		abidhb 1915	. . . . . 6 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
10	8, 9	syl 10	. . . . 5 |- (ph -> {z | A.x z e. A} = A)
11	10	opeq1d 2497	. . . 4 |- (ph -> <.{z | A.x z e. A}, {z | A.x z e. B}>. = <.A, {z | A.x z e. B}>.)
12		hbopd.3	. . . . . . 7 |- (ph -> (y e. B -> A.x y e. B))
13	12	19.21aiv 1288	. . . . . 6 |- (ph -> A.y(y e. B -> A.x y e. B))
14		abidhb 1915	. . . . . 6 |- (A.y(y e. B -> A.x y e. B) -> {z | A.x z e. B} = B)
15	13, 14	syl 10	. . . . 5 |- (ph -> {z | A.x z e. B} = B)
16	15	opeq2d 2498	. . . 4 |- (ph -> <.A, {z | A.x z e. B}>. = <.A, B>.)
17	11, 16	eqtrd 1510	. . 3 |- (ph -> <.{z | A.x z e. A}, {z | A.x z e. B}>. = <.A, B>.)
18	17	eleq2d 1544	. 2 |- (ph -> (y e. <.{z | A.x z e. A}, {z | A.x z e. B}>. <-> y e. <.A, B>.))
19		hbopd.1	. . 3 |- (ph -> A.xph)
20	19, 18	albid 1106	. 2 |- (ph -> (A.x y e. <.{z | A.x z e. A}, {z | A.x z e. B}>. <-> A.x y e. <.A, B>.))
21	6, 18, 20	3imtr3d 544	1 |- (ph -> (y e. <.A, B>. -> A.x y e. <.A, B>.))

Syntax hints:   -> wi 3  A.wal 956   = wceq 958   e. wcel 960  {cab 1466  <.cop 2415

This theorem is referenced by:  dfid3 2842

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

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