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 )