Theorem hbab1 2070

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 2068	. 2 |- ( y e. { x | ph } <-> [ y / x ] ph )
2		hbs1 1855	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3	1, 2	hbxfrbi 1401	1 |- ( y e. { x | ph } -> A. x y e. { x | ph } )

Syntax hints:   -> wi 4   A.wal 1282   e. wcel 1433   [wsb 1685   {cab 2067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-sb 1686  df-clab 2068

This theorem is referenced by:  nfsab1  2071  abeq2  2187