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| Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1824 for equivalence version. (Contributed by NM, 21-Aug-2007.) | 
| Ref | Expression | 
|---|---|
| hbxfr.1 | ⊢ 𝐴 = 𝐵 | 
| hbxfr.2 | ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| hbxfreq | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hbxfr.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
| 2 | 1 | eleq2i 2832 | . 2 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) | 
| 3 | hbxfr.2 | . 2 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) | |
| 4 | 2, 3 | hbxfrbi 1824 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1537 = wceq 1539 ∈ wcel 2107 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 df-cleq 2728 df-clel 2815 | 
| This theorem is referenced by: bnj1317 34836 bnj1441 34855 bnj1441g 34856 bnj1309 35037 | 
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