Theorem hbxfrbi 1903

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2882 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 1898	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
4	1, 2, 3	3imtr4i 282	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 197  ∀wal 1632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888

This theorem depends on definitions:  df-bi 198

This theorem is referenced by:  hbn1fw  2132  hbe1w  2136  hbe1  2179  hbexOLD  2319  hbab1  2763  hbab  2765  hbxfreq  2882  hbral  3095  bnj982  31204  bnj1095  31207  bnj1096  31208  bnj1276  31240  bnj594  31337  bnj1445  31467  bj-hbab1  33123  hbra2VD  39630