Theorem hbxfrbi 1825

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2859 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 1819	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
4	1, 2, 3	3imtr4i 292	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207

This theorem is referenced by:  hbn1fw  2046  hbe1w  2049  hbe1  2144  hbsbwOLD  2173  hbab  2718  hbabg  2719  hbxfreq  2859  hbral  3284  bnj982  34775  bnj1095  34778  bnj1096  34779  bnj1276  34811  bnj594  34909  bnj1445  35041  hbra2VD  44856