Theorem hbxfrbi 1854

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 1848	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
4	1, 2, 3	3imtr4i 295	1 ⊢ (𝜑 → ∀𝑥𝜑)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838

This proof depends on definitions:  df-bi 210

This theorem is used by:  hbn1fw  2076  hbe1w  2079  hbe1  2177  hbab  2750  hbabg  2751  hbxfreq  2892  hbral  3308  bnj982  35176  bnj1095  35179  bnj1096  35180  bnj1276  35211  bnj594  35309  bnj1445  35441  hbra2VD  45596