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Metamath Proof Explorer

Theorem bnj1049

Description: Technical lemma for bnj69 . 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
Hypotheses bnj1049.1 ⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
bnj1049.2 ⊢ ( 𝜂 ↔ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
Assertion bnj1049 ( ∀ 𝑖 ∈ 𝑛 𝜂 ↔ ∀ 𝑖 𝜂 )

Proof

Step Hyp Ref Expression
1 bnj1049.1 ⊢ ( 𝜁 ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) )
2 bnj1049.2 ⊢ ( 𝜂 ↔ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
3 df-ral ⊢ ( ∀ 𝑖 ∈ 𝑛 𝜂 ↔ ∀ 𝑖 ( 𝑖 ∈ 𝑛 → 𝜂 ) )
4 2 imbi2i ⊢ ( ( 𝑖 ∈ 𝑛 → 𝜂 ) ↔ ( 𝑖 ∈ 𝑛 → ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) ) )
5 impexp ⊢ ( ( ( 𝑖 ∈ 𝑛 ∧ ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) ) → 𝑧 ∈ 𝐵 ) ↔ ( 𝑖 ∈ 𝑛 → ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) ) )
6 4 5 bitr4i ⊢ ( ( 𝑖 ∈ 𝑛 → 𝜂 ) ↔ ( ( 𝑖 ∈ 𝑛 ∧ ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) ) → 𝑧 ∈ 𝐵 ) )
7 1 simplbi ⊢ ( 𝜁 → 𝑖 ∈ 𝑛 )
8 7 bnj708 ⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑖 ∈ 𝑛 )
9 8 pm4.71ri ⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) ↔ ( 𝑖 ∈ 𝑛 ∧ ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) ) )
10 9 bicomi ⊢ ( ( 𝑖 ∈ 𝑛 ∧ ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) ) ↔ ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) )
11 10 imbi1i ⊢ ( ( ( 𝑖 ∈ 𝑛 ∧ ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) ) → 𝑧 ∈ 𝐵 ) ↔ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
12 6 11 bitri ⊢ ( ( 𝑖 ∈ 𝑛 → 𝜂 ) ↔ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁 ) → 𝑧 ∈ 𝐵 ) )
13 12 2 bitr4i ⊢ ( ( 𝑖 ∈ 𝑛 → 𝜂 ) ↔ 𝜂 )
14 13 albii ⊢ ( ∀ 𝑖 ( 𝑖 ∈ 𝑛 → 𝜂 ) ↔ ∀ 𝑖 𝜂 )
15 3 14 bitri ⊢ ( ∀ 𝑖 ∈ 𝑛 𝜂 ↔ ∀ 𝑖 𝜂 )