Theorem hbeld 1921

Description: Deduction version of bound-variable hypothesis builder hbel 1573.

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 1009	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
2	1	hbab 1474	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
3		hba1 1009	. . . . 5 |- (A.x z e. B -> A.xA.x z e. B)
4	3	hbab 1474	. . . 4 |- (y e. {z | A.x z e. B} -> A.x y e. {z | A.x z e. B})
5	2, 4	hbel 1573	. . 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 1292	. . . 4 |- (ph -> A.y(y e. A -> A.x y e. A))
9		abidhb 1919	. . . 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 1292	. . . 4 |- (ph -> A.y(y e. B -> A.x y e. B))
13		abidhb 1919	. . . 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 1549	. 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 1110	. 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 545	1 |- (ph -> (A e. B -> A.x A e. B))

Syntax hints:   -> wi 3  A.wal 958   = wceq 960   e. wcel 962  {cab 1470

This theorem is referenced by:  hbsbc1gd 1991  hbsbcgd 1992  hbcsb1gd 2036  hbcsbgd 2037

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1178  df-clab 1471  df-cleq 1476  df-clel 1479  df-v 1819

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