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	⊢ (φ ↔ ψ)
hbxfrbi.2	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
hbxfrbi	⊢ (φ → ∀xφ)

Proof of Theorem hbxfrbi

Step	Hyp	Ref	Expression
1		hbxfrbi.2	. 2 ⊢ (ψ → ∀xψ)
2		hbxfrbi.1	. 2 ⊢ (φ ↔ ψ)
3	2	albii 1566	. 2 ⊢ (∀xφ ↔ ∀xψ)
4	1, 2, 3	3imtr4i 257	1 ⊢ (φ → ∀xφ)

Syntax hints:   → wi 4   ↔ wb 176  ∀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