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Theorem bnj1047 31420
Description: Technical lemma for bnj69 31457. 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 327 . . 3 ((𝑗 E 𝑖𝜂′) ↔ (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
43ralbii 3126 . 2 (∀𝑗𝑛 (𝑗 E 𝑖𝜂′) ↔ ∀𝑗𝑛 (𝑗 E 𝑖[𝑗 / 𝑖]𝜂))
51, 4bitr4i 269 1 (𝜌 ↔ ∀𝑗𝑛 (𝑗 E 𝑖𝜂′))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wral 3054  [wsbc 3595   class class class wbr 4808   E cep 5188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904
This theorem depends on definitions:  df-bi 198  df-ral 3059
This theorem is referenced by: (None)
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