Theorem hbopd 2493

Description: Deduction version of bound-variable hypothesis builder hbop 2492.

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 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. B -> A.xA.x z e. B)
4	3	hbab 1465	. . . 4 |- (y e. {z | A.x z e. B} -> A.x y e. {z | A.x z e. B})
5	2, 4	hbop 2492	. . 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 1284	. . . . . 6 |- (ph -> A.y(y e. A -> A.x y e. A))
9		abidhb 1908	. . . . . 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 2489	. . . 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 1284	. . . . . 6 |- (ph -> A.y(y e. B -> A.x y e. B))
14		abidhb 1908	. . . . . 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 2490	. . . 4 |- (ph -> <.A, {z | A.x z e. B}>. = <.A, B>.)
17	11, 16	eqtrd 1504	. . 3 |- (ph -> <.{z | A.x z e. A}, {z | A.x z e. B}>. = <.A, B>.)
18	17	eleq2d 1538	. 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 1102	. 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 541	1 |- (ph -> (y e. <.A, B>. -> A.x y e. <.A, B>.))

Syntax hints:   -> wi 3  A.wal 952   = wceq 954   e. wcel 956  {cab 1461  <.cop 2407

This theorem is referenced by:  dfid3 2831

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-12 966  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409  df-op 2412

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