Theorem hbrab 1770

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 1649	. 2 |- {y e. A | ph} = {y | (y e. A /\ ph)}
2		ax-17 969	. . . . 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 1563	. . . 4 |- (y e. A -> A.x y e. A)
5		hbrab.1	. . . 4 |- (ph -> A.xph)
6	4, 5	hban 1007	. . 3 |- ((y e. A /\ ph) -> A.x(y e. A /\ ph))
7	6	hbab 1465	. 2 |- (z e. {y | (y e. A /\ ph)} -> A.x z e. {y | (y e. A /\ ph)})
8	1, 7	hbxfr 1560	1 |- (z e. {y e. A | ph} -> A.x z e. {y e. A | ph})

Syntax hints:   -> wi 3   /\ wa 223  A.wal 952   e. wcel 956  {cab 1461  {crab 1645

This theorem is referenced by:  scottex 4696  lble 6002  fgsb 10480  fgsb2 10485

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-rab 1649

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