Theorem hbab 2220

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995.)

Hypothesis

Ref	Expression
hbab.1	|- ( ph -> A. x ph )

Assertion

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

Distinct variable group:   x, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem hbab

Step	Hyp	Ref	Expression
1		df-clab 2216	. 2 |- ( z e. { y | ph } <-> [ z / y ] ph )
2		hbab.1	. . 3 |- ( ph -> A. x ph )
3	2	hbsb 2000	. 2 |- ( [ z / y ] ph -> A. x [ z / y ] ph )
4	1, 3	hbxfrbi 1518	1 |- ( z e. { y | ph } -> A. x z e. { y | ph } )

Syntax hints:   -> wi 4   A.wal 1393   [wsb 1808   e. wcel 2200   {cab 2215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216

This theorem is referenced by:  nfsab  2221