Theorem hbab 2128

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 2124	. 2 |- ( z e. { y | ph } <-> [ z / y ] ph )
2		hbab.1	. . 3 |- ( ph -> A. x ph )
3	2	hbsb 1920	. 2 |- ( [ z / y ] ph -> A. x [ z / y ] ph )
4	1, 3	hbxfrbi 1448	1 |- ( z e. { y | ph } -> A. x z e. { y | ph } )

Syntax hints:   -> wi 4   A.wal 1329   e. wcel 1480   [wsb 1735   {cab 2123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124

This theorem is referenced by:  nfsab  2129