Theorem hbeqd 1909

Description: Deduction version of bound-variable hypothesis builder hbeq 1562.

Hypotheses

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

Assertion

Ref	Expression
hbeqd	|- (ph -> (A = B -> A.x A = B))

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

Proof of Theorem hbeqd

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	hbeq 1562	. . 3 |- ({z | A.x z e. A} = {z | A.x z e. B} -> A.x{z | A.x z e. A} = {z | A.x z e. B})
6	5	a1i 8	. 2 |- (ph -> ({z | A.x z e. A} = {z | A.x z e. B} -> A.x{z | A.x z e. A} = {z | A.x z e. B}))
7		hbeqd.2	. . . . 5 |- (ph -> (y e. A -> A.x y e. A))
8	7	19.21aiv 1284	. . . 4 |- (ph -> A.y(y e. A -> A.x y e. A))
9		abidhb 1908	. . . 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		hbeqd.3	. . . . 5 |- (ph -> (y e. B -> A.x y e. B))
12	11	19.21aiv 1284	. . . 4 |- (ph -> A.y(y e. B -> A.x y e. B))
13		abidhb 1908	. . . 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	eqeq12d 1486	. 2 |- (ph -> ({z | A.x z e. A} = {z | A.x z e. B} <-> A = B))
16		hbeqd.1	. . 3 |- (ph -> A.xph)
17	16, 15	albid 1102	. 2 |- (ph -> (A.x{z | A.x z e. A} = {z | A.x z e. B} <-> A.x A = B))
18	6, 15, 17	3imtr3d 541	1 |- (ph -> (A = B -> A.x A = B))

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

This theorem is referenced by:  sbcralt 1986  sbcralgf 1988  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-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808

Copyright terms: Public domain