Theorem hbeld 1952

Description: Deduction version of bound-variable hypothesis builder hbel 1603.

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 1035	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
2	1	hbab 1503	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
3		hba1 1035	. . . . 5 |- (A.x z e. B -> A.xA.x z e. B)
4	3	hbab 1503	. . . 4 |- (y e. {z | A.x z e. B} -> A.x y e. {z | A.x z e. B})
5	2, 4	hbel 1603	. . 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 1319	. . . 4 |- (ph -> A.y(y e. A -> A.x y e. A))
9		abidhb 1950	. . . 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 1319	. . . 4 |- (ph -> A.y(y e. B -> A.x y e. B))
13		abidhb 1950	. . . 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 1579	. 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 1136	. 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 544	1 |- (ph -> (A e. B -> A.x A e. B))

Syntax hints:   -> wi 3  A.wal 986   = wceq 988   e. wcel 990  {cab 1499

This theorem is referenced by:  hbsbc1gd 2023  hbsbcgd 2024  hbcsb1gd 2070  hbcsbgd 2071

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-12 1000  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1013  df-sb 1205  df-clab 1500  df-cleq 1505  df-clel 1508  df-v 1850

Copyright terms: Public domain