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Theorem bnj1047 31379
 Description: Technical lemma for bnj69 31416. 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 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
bnj1047.2 (𝜂′[𝑗 / 𝑖]𝜂)
Assertion
Ref Expression
bnj1047 (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖𝜂′))

Proof of Theorem bnj1047
StepHypRef Expression
1 bnj1047.1 . 2 (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
2 bnj1047.2 . . . 4 (𝜂′[𝑗 / 𝑖]𝜂)
32imbi2i 325 . . 3 ((𝑗 E 𝑖𝜂′) ↔ (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
43ralbii 3129 . 2 (∀𝑗𝑛 (𝑗 E 𝑖𝜂′) ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
51, 4bitr4i 267 1 (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖𝜂′))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wral 3061  [wsbc 3587   class class class wbr 4787   E cep 5162 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885 This theorem depends on definitions:  df-bi 197  df-ral 3066 This theorem is referenced by: (None)
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