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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 1742 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 2680 | . 2 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) |
3 | hbxfr.2 | . 2 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) | |
4 | 2, 3 | hbxfrbi 1742 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 = wceq 1475 ∈ wcel 1977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-cleq 2603 df-clel 2606 |
This theorem is referenced by: bnj1317 30146 bnj1441 30165 bnj1309 30344 |
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