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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 1816 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 2901 | . 2 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) |
3 | hbxfr.2 | . 2 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) | |
4 | 2, 3 | hbxfrbi 1816 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 = wceq 1528 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-cleq 2811 df-clel 2890 |
This theorem is referenced by: bnj1317 31992 bnj1441 32011 bnj1441g 32012 bnj1309 32191 |
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