Theorem hbrab 1773

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 groups:   x,y,z   z,A

Proof of Theorem hbrab

Step	Hyp	Ref	Expression
1		df-rab 1652	. 2 |- {y e. A | ph} = {y | (y e. A /\ ph)}
2		ax-17 971	. . . . 5 |- (z e. y -> A.x z e. y)
3		hbrab.2	. . . . 5 |- (z e. A -> A.x z e. A)
4	2, 3	hbel 1566	. . . 4 |- (y e. A -> A.x y e. A)
5		hbrab.1	. . . 4 |- (ph -> A.xph)
6	4, 5	hban 1009	. . 3 |- ((y e. A /\ ph) -> A.x(y e. A /\ ph))
7	6	hbab 1467	. 2 |- (z e. {y | (y e. A /\ ph)} -> A.x z e. {y | (y e. A /\ ph)})
8	1, 7	hbxfr 1563	1 |- (z e. {y e. A | ph} -> A.x z e. {y e. A | ph})

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   e. wcel 958  {cab 1463  {crab 1648

This theorem is referenced by:  scottex 4716  lble 6047  fgsb 10570  fgsbOLD 10571  fgsb2 10580

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

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