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Theorem bnj1039 31419
Description: Technical lemma for bnj69 31458. 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 3353 . . 3 𝑗 ∈ V
3 bnj1039.1 . . . . 5 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4 nfra1 3088 . . . . 5 𝑖𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅))
53, 4nfxfr 1948 . . . 4 𝑖𝜓
65sbcgf 3660 . . 3 (𝑗 ∈ V → ([𝑗 / 𝑖]𝜓𝜓))
72, 6ax-mp 5 . 2 ([𝑗 / 𝑖]𝜓𝜓)
81, 7, 33bitri 288 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 197   = wceq 1652  wcel 2155  wral 3055  Vcvv 3350  [wsbc 3596   ciun 4676  suc csuc 5910  cfv 6068  ωcom 7263   predc-bnj14 31137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-ral 3060  df-v 3352  df-sbc 3597
This theorem is referenced by:  bnj1128  31438
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