Theorem hbxfrbi 1568

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2457 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
hbxfrbi.1	|- (ph <-> ps)
hbxfrbi.2	|- (ps -> A.xps)

Assertion

Ref	Expression
hbxfrbi	|- (ph -> A.xph)

Proof of Theorem hbxfrbi

Step	Hyp	Ref	Expression
1		hbxfrbi.2	. 2 |- (ps -> A.xps)
2		hbxfrbi.1	. 2 |- (ph <-> ps)
3	2	albii 1566	. 2 |- (A.xph <-> A.xps)
4	1, 2, 3	3imtr4i 257	1 |- (ph -> A.xph)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  hbe1w  1708  hbe1  1731  hbanOLD  1829  hb3anOLD  1831  hbex  1841  hbab1  2342  hbab  2344  cleqh  2450  hbxfreq  2457  hbral  2663  hbra1  2664