Theorem hbxfreq 2277

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1465 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 2237	. 2 |- ( y e. A <-> y e. B )
3		hbxfr.2	. 2 |- ( y e. B -> A. x y e. B )
4	2, 3	hbxfrbi 1465	1 |- ( y e. A -> A. x y e. A )

Syntax hints:   -> wi 4   A.wal 1346   = wceq 1348   e. wcel 2141

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-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-cleq 2163  df-clel 2166

This theorem is referenced by: (None)