Theorem hbxfrbi 1749

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2727 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 1744	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
4	1, 2, 3	3imtr4i 281	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734

This theorem depends on definitions:  df-bi 197

This theorem is referenced by:  hbn1fw  1969  hbe1w  1973  hbe1  2018  hbexOLD  2149  hbab1  2610  hbab  2612  hbxfreq  2727  hbral  2938  bnj982  30554  bnj1095  30557  bnj1096  30558  bnj1276  30590  bnj594  30687  bnj1445  30817  bj-hbab1  32411  hbra2VD  38576