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Theorem bnj1039 30293
Description: Technical lemma for bnj69 30332. 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
bnj1039.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1039.2 (𝜓′[𝑗 / 𝑖]𝜓)
Assertion
Ref Expression
bnj1039 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))

Proof of Theorem bnj1039
StepHypRef Expression
1 bnj1039.2 . 2 (𝜓′[𝑗 / 𝑖]𝜓)
2 vex 3176 . . 3 𝑗 ∈ V
3 bnj1039.1 . . . . 5 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4 nfra1 2925 . . . . 5 𝑖𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
53, 4nfxfr 1771 . . . 4 𝑖𝜓
65sbcgf 3468 . . 3 (𝑗 ∈ V → ([𝑗 / 𝑖]𝜓𝜓))
72, 6ax-mp 5 . 2 ([𝑗 / 𝑖]𝜓𝜓)
81, 7, 33bitri 285 1 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  [wsbc 3402   ciun 4455  suc csuc 5642  cfv 5804  ωcom 6957   predc-bnj14 30007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-v 3175  df-sbc 3403
This theorem is referenced by:  bnj1128  30312
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