Theorem hbxfrbi 1821

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2942 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 1816	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
4	1, 2, 3	3imtr4i 294	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 209

This theorem is referenced by:  hbn1fw  2048  hbe1w  2051  hbe1  2143  hbab1  2807  hbab  2810  hbabg  2811  hbxfreq  2942  hbral  3221  bnj982  32045  bnj1095  32048  bnj1096  32049  bnj1276  32081  bnj594  32179  bnj1445  32311  hbra2VD  41187