Theorem bnj1047 32853

Description: Technical lemma for bnj69 32890. 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.)

Hypotheses

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

Assertion

Ref	Expression
bnj1047	⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → 𝜂′))

Proof of Theorem bnj1047

Step	Hyp	Ref	Expression
1		bnj1047.1	. 2 ⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂))
2		bnj1047.2	. . . 4 ⊢ (𝜂′ ↔ [𝑗 / 𝑖]𝜂)
3	2	imbi2i 335	. . 3 ⊢ ((𝑗 E 𝑖 → 𝜂′) ↔ (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂))
4	3	ralbii 3090	. 2 ⊢ (∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → 𝜂′) ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂))
5	1, 4	bitr4i 277	1 ⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → 𝜂′))

Syntax hints:   → wi 4   ↔ wb 205  ∀wral 3063  [wsbc 3711   class class class wbr 5070   E cep 5485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-ral 3068

This theorem is referenced by: (None)