Theorem hbeld 1914

Description: Deduction version of bound-variable hypothesis builder hbel 1566.

Hypotheses

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

Assertion

Ref	Expression
hbeld	|- (ph -> (A e. B -> A.x A e. B))

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

Proof of Theorem hbeld

Step	Hyp	Ref	Expression
1		hba1 1003	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
2	1	hbab 1467	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
3		hba1 1003	. . . . 5 |- (A.x z e. B -> A.xA.x z e. B)
4	3	hbab 1467	. . . 4 |- (y e. {z | A.x z e. B} -> A.x y e. {z | A.x z e. B})
5	2, 4	hbel 1566	. . 3 |- ({z | A.x z e. A} e. {z | A.x z e. B} -> A.x{z | A.x z e. A} e. {z | A.x z e. B})
6	5	a1i 8	. 2 |- (ph -> ({z | A.x z e. A} e. {z | A.x z e. B} -> A.x{z | A.x z e. A} e. {z | A.x z e. B}))
7		hbeld.2	. . . . 5 |- (ph -> (y e. A -> A.x y e. A))
8	7	19.21aiv 1286	. . . 4 |- (ph -> A.y(y e. A -> A.x y e. A))
9		abidhb 1912	. . . 4 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
10	8, 9	syl 10	. . 3 |- (ph -> {z | A.x z e. A} = A)
11		hbeld.3	. . . . 5 |- (ph -> (y e. B -> A.x y e. B))
12	11	19.21aiv 1286	. . . 4 |- (ph -> A.y(y e. B -> A.x y e. B))
13		abidhb 1912	. . . 4 |- (A.y(y e. B -> A.x y e. B) -> {z | A.x z e. B} = B)
14	12, 13	syl 10	. . 3 |- (ph -> {z | A.x z e. B} = B)
15	10, 14	eleq12d 1542	. 2 |- (ph -> ({z | A.x z e. A} e. {z | A.x z e. B} <-> A e. B))
16		hbeld.1	. . 3 |- (ph -> A.xph)
17	16, 15	albid 1104	. 2 |- (ph -> (A.x{z | A.x z e. A} e. {z | A.x z e. B} <-> A.x A e. B))
18	6, 15, 17	3imtr3d 542	1 |- (ph -> (A e. B -> A.x A e. B))

Syntax hints:   -> wi 3  A.wal 954   = wceq 956   e. wcel 958  {cab 1463

This theorem is referenced by:  hbsbc1gd 1983  hbsbcgd 1984  hbcsb1gd 2027  hbcsbgd 2028

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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