Theorem hbab1 2182

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 2180	. 2 |- ( y e. { x | ph } <-> [ y / x ] ph )
2		hbs1 1954	. 2 |- ( [ y / x ] ph -> A. x [ y / x ] ph )
3	1, 2	hbxfrbi 1483	1 |- ( y e. { x | ph } -> A. x y e. { x | ph } )

Syntax hints:   -> wi 4   A.wal 1362   [wsb 1773   e. wcel 2164   {cab 2179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2180

This theorem is referenced by:  nfsab1  2183  abeq2  2302