Theorem hbxfrbi 1742

Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2717 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 1737	. 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
4	1, 2, 3	3imtr4i 280	1 ⊢ (𝜑 → ∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728

This theorem depends on definitions:  df-bi 196

This theorem is referenced by:  hbn1fw  1959  hbe1w  1963  hbe1  2008  hbexOLD  2138  hbab1  2599  hbab  2601  hbxfreq  2717  hbral  2927  bnj982  30103  bnj1095  30106  bnj1096  30107  bnj1276  30139  bnj594  30236  bnj1445  30366  bj-hbab1  31959  hbra2VD  38118