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Mirrors > Home > ILE Home > Th. List > hbxfreq | GIF version |
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1401 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 2145 | . 2 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) |
3 | hbxfr.2 | . 2 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) | |
4 | 2, 3 | hbxfrbi 1401 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 = wceq 1284 ∈ wcel 1433 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-4 1440 ax-17 1459 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 df-clel 2077 |
This theorem is referenced by: (None) |
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