Theorem hbab1 2029

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 2027	. 2 |- ( y e. { x | ph } <-> [ y / x ] ph )
2		hbs1 1814	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3	1, 2	hbxfrbi 1361	1 |- ( y e. { x | ph } -> A. x y e. { x | ph } )

Syntax hints:   -> wi 4   A.wal 1241   e. wcel 1393   [wsb 1645   {cab 2026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027

This theorem is referenced by:  nfsab1  2030  abeq2  2146