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Theorem bnj591 31795
 Description: Technical lemma for bnj852 31805. 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 3763 . 2 ([𝑘 / 𝑗]𝜃[𝑘 / 𝑗]((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
3 vex 3443 . . 3 𝑘 ∈ V
4 fveq2 6545 . . . . 5 (𝑗 = 𝑘 → (𝑓𝑗) = (𝑓𝑘))
5 fveq2 6545 . . . . 5 (𝑗 = 𝑘 → (𝑔𝑗) = (𝑔𝑘))
64, 5eqeq12d 2812 . . . 4 (𝑗 = 𝑘 → ((𝑓𝑗) = (𝑔𝑗) ↔ (𝑓𝑘) = (𝑔𝑘)))
76imbi2d 342 . . 3 (𝑗 = 𝑘 → (((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘))))
83, 7sbcie 3746 . 2 ([𝑘 / 𝑗]((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))
92, 8bitri 276 1 ([𝑘 / 𝑗]𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083  [wsbc 3711  ‘cfv 6232 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-iota 6196  df-fv 6240 This theorem is referenced by:  bnj580  31797
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