Theorem hbxfrbi 1827

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2869 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 1822	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
4	1, 2, 3	3imtr4i 292	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812

This theorem depends on definitions:  df-bi 206

This theorem is referenced by:  hbn1fw  2048  hbe1w  2051  hbe1  2139  hbsbw  2169  hbab1OLD  2725  hbab  2726  hbabg  2727  hbxfreq  2869  hbral  3146  bnj982  32758  bnj1095  32761  bnj1096  32762  bnj1276  32794  bnj594  32892  bnj1445  33024  hbra2VD  42480