Theorem hbrab 1311

Description: A variable not free in a wff remains so in a restricted class abstraction.

Hypotheses

Ref	Expression
hbrab.1	|- (ph -> A.xph)
hbrab.2	|- (z e. A -> A.x z e. A)

Assertion

Ref	Expression
hbrab	|- (z e. {y e. A | ph} -> A.x z e. {y e. A | ph})

Distinct variable group(s):   x,y,z   z,A

Proof of Theorem hbrab

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 |- (z e. y -> A.x z e. y)
2		hbrab.2	. . . . 5 |- (z e. A -> A.x z e. A)
3	1, 2	hbel 1172	. . . 4 |- (y e. A -> A.x y e. A)
4		hbrab.1	. . . 4 |- (ph -> A.xph)
5	3, 4	hban 704	. . 3 |- ((y e. A /\ ph) -> A.x(y e. A /\ ph))
6	5	hbab 1096	. 2 |- (z e. {y | (y e. A /\ ph)} -> A.x z e. {y | (y e. A /\ ph)})
7		df-rab 1208	. . 3 |- {y e. A | ph} = {y | (y e. A /\ ph)}
8	7	eleq2i 1153	. 2 |- (z e. {y e. A | ph} <-> z e. {y | (y e. A /\ ph)})
9	8	bial 695	. 2 |- (A.x z e. {y e. A | ph} <-> A.x z e. {y | (y e. A /\ ph)})
10	6, 8, 9	3imtr4 192	1 |- (z e. {y e. A | ph} -> A.x z e. {y e. A | ph})

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672   e. wel 803  {cab 1090   e. wcel 1092  {crab 1204

This theorem is referenced by:  scottex 3541

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rab 1208

metamath.org