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Mirrors > Home > MPE Home > Th. List > hbxfreq | Structured version Visualization version GIF version |
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1822 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 2831 | . 2 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) |
3 | hbxfr.2 | . 2 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) | |
4 | 2, 3 | hbxfrbi 1822 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 = wceq 1537 ∈ wcel 2106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-cleq 2727 df-clel 2814 |
This theorem is referenced by: bnj1317 34814 bnj1441 34833 bnj1441g 34834 bnj1309 35015 |
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