Theorem hbab 2452

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 2448	. 2 |- ( z e. { y | ph } <-> [ z / y ] ph )
2		hbab.1	. . 3 |- ( ph -> A. x ph )
3	2	hbsb 2280	. 2 |- ( [ z / y ] ph -> A. x [ z / y ] ph )
4	1, 3	hbxfrbi 1704	1 |- ( z e. { y | ph } -> A. x z e. { y | ph } )

Syntax hints:   -> wi 4   A.wal 1452   [wsb 1807   e. wcel 1897   {cab 2447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448

This theorem is referenced by:  nfsab  2453  bnj1441  29700  bnj1309  29879