Theorem hbxfrbi 1858

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2892 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 1852	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
4	1, 2, 3	3imtr4i 295	1 ⊢ (𝜑 → ∀𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210

This theorem is used by:  hbn1fw  2080  hbe1w  2083  hbe1  2180  hbab  2750  hbabg  2751  hbxfreq  2892  hbral  3308  bnj982  35275  bnj1095  35278  bnj1096  35279  bnj1276  35310  bnj594  35408  bnj1445  35540  hbra2VD  45669