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Theorem bnj591 34756
Description: Technical lemma for bnj852 34766. 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.)
Hypothesis
Ref Expression
bnj591.1 (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
Assertion
Ref Expression
bnj591 ([𝑘 / 𝑗]𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))
Distinct variable groups:   𝐷,𝑗   𝜒,𝑗   𝑗,𝜒′   𝑓,𝑗   𝑔,𝑗   𝑗,𝑘   𝑗,𝑛
Allowed substitution hints:   𝜒(𝑓,𝑔,𝑘,𝑛)   𝜃(𝑓,𝑔,𝑗,𝑘,𝑛)   𝐷(𝑓,𝑔,𝑘,𝑛)   𝜒′(𝑓,𝑔,𝑘,𝑛)

Proof of Theorem bnj591
StepHypRef Expression
1 bnj591.1 . . 3 (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
21sbcbii 3837 . 2 ([𝑘 / 𝑗]𝜃[𝑘 / 𝑗]((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
3 vex 3466 . . 3 𝑘 ∈ V
4 fveq2 6901 . . . . 5 (𝑗 = 𝑘 → (𝑓𝑗) = (𝑓𝑘))
5 fveq2 6901 . . . . 5 (𝑗 = 𝑘 → (𝑔𝑗) = (𝑔𝑘))
64, 5eqeq12d 2742 . . . 4 (𝑗 = 𝑘 → ((𝑓𝑗) = (𝑔𝑗) ↔ (𝑓𝑘) = (𝑔𝑘)))
76imbi2d 339 . . 3 (𝑗 = 𝑘 → (((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘))))
83, 7sbcie 3820 . 2 ([𝑘 / 𝑗]((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))
92, 8bitri 274 1 ([𝑘 / 𝑗]𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1534  wcel 2099  [wsbc 3776  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-iota 6506  df-fv 6562
This theorem is referenced by:  bnj580  34758
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