Theorem cbvrab 1910

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions.

Hypotheses

Ref	Expression
cbvrab.1	|- (z e. A -> A.x z e. A)
cbvrab.2	|- (z e. A -> A.y z e. A)
cbvrab.3	|- (ph -> A.yph)
cbvrab.4	|- (ps -> A.xps)
cbvrab.5	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvrab	|- {x e. A | ph} = {y e. A | ps}

Distinct variable groups:   x,y,z   z,A

Proof of Theorem cbvrab

Step	Hyp	Ref	Expression
1		ax-17 971	. . . . 5 |- (z e. x -> A.y z e. x)
2		cbvrab.2	. . . . 5 |- (z e. A -> A.y z e. A)
3	1, 2	hbel 1566	. . . 4 |- (x e. A -> A.y x e. A)
4		cbvrab.3	. . . 4 |- (ph -> A.yph)
5	3, 4	hban 1009	. . 3 |- ((x e. A /\ ph) -> A.y(x e. A /\ ph))
6		ax-17 971	. . . . 5 |- (z e. y -> A.x z e. y)
7		cbvrab.1	. . . . 5 |- (z e. A -> A.x z e. A)
8	6, 7	hbel 1566	. . . 4 |- (y e. A -> A.x y e. A)
9		cbvrab.4	. . . 4 |- (ps -> A.xps)
10	8, 9	hban 1009	. . 3 |- ((y e. A /\ ps) -> A.x(y e. A /\ ps))
11		eleq1 1534	. . . 4 |- (x = y -> (x e. A <-> y e. A))
12		cbvrab.5	. . . 4 |- (x = y -> (ph <-> ps))
13	11, 12	anbi12d 628	. . 3 |- (x = y -> ((x e. A /\ ph) <-> (y e. A /\ ps)))
14	5, 10, 13	cbvab 1908	. 2 |- {x | (x e. A /\ ph)} = {y | (y e. A /\ ps)}
15		df-rab 1652	. 2 |- {x e. A | ph} = {x | (x e. A /\ ph)}
16		df-rab 1652	. 2 |- {y e. A | ps} = {y | (y e. A /\ ps)}
17	14, 15, 16	3eqtr4 1505	1 |- {x e. A | ph} = {y e. A | ps}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  {cab 1463  {crab 1648

This theorem is referenced by:  cbvrabv 1911  elrabsf 1963  iunrab 2596  scottexs 4718  scott0s 4719  hta 4728

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-rab 1652  df-v 1812

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