Theorem hbxfreq 2730

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1752 for equivalence version. (Contributed by NM, 21-Aug-2007.)

Hypotheses

Ref	Expression
hbxfr.1	|- A = B
hbxfr.2	|- ( y e. B -> A. x y e. B )

Assertion

Ref	Expression
hbxfreq	|- ( y e. A -> A. x y e. A )

Proof of Theorem hbxfreq

Step	Hyp	Ref	Expression
1		hbxfr.1	. . 3 |- A = B
2	1	eleq2i 2693	. 2 |- ( y e. A <-> y e. B )
3		hbxfr.2	. 2 |- ( y e. B -> A. x y e. B )
4	2, 3	hbxfrbi 1752	1 |- ( y e. A -> A. x y e. A )

Syntax hints:   -> wi 4   A.wal 1481   = wceq 1483   e. wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618

This theorem is referenced by:  bnj1317  30892  bnj1441  30911  bnj1309  31090