Theorem hbab1 2159

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 2157	. 2 |- ( y e. { x | ph } <-> [ y / x ] ph )
2		hbs1 1931	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3	1, 2	hbxfrbi 1465	1 |- ( y e. { x | ph } -> A. x y e. { x | ph } )

Syntax hints:   -> wi 4   A.wal 1346   [wsb 1755   e. wcel 2141   {cab 2156

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157

This theorem is referenced by:  nfsab1  2160  abeq2  2279