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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 1752 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 2693 | . 2 ⊢ (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵) |
3 | hbxfr.2 | . 2 ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵) | |
4 | 2, 3 | hbxfrbi 1752 | 1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 = wceq 1483 ∈ wcel 1990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 df-cleq 2615 df-clel 2618 |
This theorem is referenced by: bnj1317 30892 bnj1441 30911 bnj1309 31090 |
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