Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj591 Structured version   Visualization version   GIF version

Theorem bnj591 34925
Description: Technical lemma for bnj852 34935. 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 3846 . 2 ([𝑘 / 𝑗]𝜃[𝑘 / 𝑗]((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
3 vex 3484 . . 3 𝑘 ∈ V
4 fveq2 6906 . . . . 5 (𝑗 = 𝑘 → (𝑓𝑗) = (𝑓𝑘))
5 fveq2 6906 . . . . 5 (𝑗 = 𝑘 → (𝑔𝑗) = (𝑔𝑘))
64, 5eqeq12d 2753 . . . 4 (𝑗 = 𝑘 → ((𝑓𝑗) = (𝑔𝑗) ↔ (𝑓𝑘) = (𝑔𝑘)))
76imbi2d 340 . . 3 (𝑗 = 𝑘 → (((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘))))
83, 7sbcie 3830 . 2 ([𝑘 / 𝑗]((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))
92, 8bitri 275 1 ([𝑘 / 𝑗]𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  [wsbc 3788  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569
This theorem is referenced by:  bnj580  34927
  Copyright terms: Public domain W3C validator