Theorem bnj1047

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	bnj1047.1	⊢ ( 𝜌 ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] 𝜂 ) )
	bnj1047.2	⊢ ( 𝜂′ ↔ [ 𝑗 / 𝑖 ] 𝜂 )
Assertion	bnj1047	⊢ ( 𝜌 ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → 𝜂′ ) )

Proof

Step	Hyp	Ref	Expression
1		bnj1047.1	⊢ ( 𝜌 ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] 𝜂 ) )
2		bnj1047.2	⊢ ( 𝜂′ ↔ [ 𝑗 / 𝑖 ] 𝜂 )
3	2	imbi2i	⊢ ( ( 𝑗 E 𝑖 → 𝜂′ ) ↔ ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] 𝜂 ) )
4	3	ralbii	⊢ ( ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → 𝜂′ ) ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] 𝜂 ) )
5	1 4	bitr4i	⊢ ( 𝜌 ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → 𝜂′ ) )