Theorem hbxfreq 2144

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1361 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 2104	. 2 |- ( y e. A <-> y e. B )
3		hbxfr.2	. 2 |- ( y e. B -> A. x y e. B )
4	2, 3	hbxfrbi 1361	1 |- ( y e. A -> A. x y e. A )

Syntax hints:   -> wi 4   A.wal 1241   = wceq 1243   e. wcel 1393

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-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036

This theorem is referenced by: (None)