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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1049 | Structured version Visualization version GIF version |
Description: Technical lemma for bnj69 32986. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1049.1 | ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) |
bnj1049.2 | ⊢ (𝜂 ↔ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)) |
Ref | Expression |
---|---|
bnj1049 | ⊢ (∀𝑖 ∈ 𝑛 𝜂 ↔ ∀𝑖𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3071 | . 2 ⊢ (∀𝑖 ∈ 𝑛 𝜂 ↔ ∀𝑖(𝑖 ∈ 𝑛 → 𝜂)) | |
2 | bnj1049.2 | . . . . . . 7 ⊢ (𝜂 ↔ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)) | |
3 | 2 | imbi2i 336 | . . . . . 6 ⊢ ((𝑖 ∈ 𝑛 → 𝜂) ↔ (𝑖 ∈ 𝑛 → ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))) |
4 | impexp 451 | . . . . . 6 ⊢ (((𝑖 ∈ 𝑛 ∧ (𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁)) → 𝑧 ∈ 𝐵) ↔ (𝑖 ∈ 𝑛 → ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))) | |
5 | 3, 4 | bitr4i 277 | . . . . 5 ⊢ ((𝑖 ∈ 𝑛 → 𝜂) ↔ ((𝑖 ∈ 𝑛 ∧ (𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁)) → 𝑧 ∈ 𝐵)) |
6 | bnj1049.1 | . . . . . . . . . 10 ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) | |
7 | 6 | simplbi 498 | . . . . . . . . 9 ⊢ (𝜁 → 𝑖 ∈ 𝑛) |
8 | 7 | bnj708 32732 | . . . . . . . 8 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑖 ∈ 𝑛) |
9 | 8 | pm4.71ri 561 | . . . . . . 7 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) ↔ (𝑖 ∈ 𝑛 ∧ (𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁))) |
10 | 9 | bicomi 223 | . . . . . 6 ⊢ ((𝑖 ∈ 𝑛 ∧ (𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁)) ↔ (𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁)) |
11 | 10 | imbi1i 350 | . . . . 5 ⊢ (((𝑖 ∈ 𝑛 ∧ (𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁)) → 𝑧 ∈ 𝐵) ↔ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)) |
12 | 5, 11 | bitri 274 | . . . 4 ⊢ ((𝑖 ∈ 𝑛 → 𝜂) ↔ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)) |
13 | 12, 2 | bitr4i 277 | . . 3 ⊢ ((𝑖 ∈ 𝑛 → 𝜂) ↔ 𝜂) |
14 | 13 | albii 1826 | . 2 ⊢ (∀𝑖(𝑖 ∈ 𝑛 → 𝜂) ↔ ∀𝑖𝜂) |
15 | 1, 14 | bitri 274 | 1 ⊢ (∀𝑖 ∈ 𝑛 𝜂 ↔ ∀𝑖𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1540 ∈ wcel 2110 ∀wral 3066 ‘cfv 6432 ∧ w-bnj17 32661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ral 3071 df-bnj17 32662 |
This theorem is referenced by: bnj1052 32951 |
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