Theorem hbab1 2128

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
hbab1	|- ( y e. { x | ph } -> A. x y e. { x | ph } )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem hbab1

Step	Hyp	Ref	Expression
1		df-clab 2126	. 2 |- ( y e. { x | ph } <-> [ y / x ] ph )
2		hbs1 1911	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3	1, 2	hbxfrbi 1448	1 |- ( y e. { x | ph } -> A. x y e. { x | ph } )

Syntax hints:   -> wi 4   A.wal 1329   e. wcel 1480   [wsb 1735   {cab 2125

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-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514

This theorem depends on definitions:  df-bi 116  df-sb 1736  df-clab 2126

This theorem is referenced by:  nfsab1  2129  abeq2  2248