Theorem hbxfrbi 1847

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2894 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 1841	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
4	1, 2, 3	3imtr4i 294	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831

This theorem depends on definitions:  df-bi 209

This theorem is referenced by:  hbn1fw  2069  hbe1w  2072  hbe1  2179  hbab  2752  hbabg  2753  hbxfreq  2894  hbral  3308  bnj982  35076  bnj1095  35079  bnj1096  35080  bnj1276  35111  bnj594  35209  bnj1445  35341  hbra2VD  45440