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Theorem bnj126 32032
Description: Technical lemma for bnj150 32035. 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
bnj126.1 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj126.2 (𝜓′[1o / 𝑛]𝜓)
bnj126.3 (𝜓″[𝐹 / 𝑓]𝜓′)
bnj126.4 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
Assertion
Ref Expression
bnj126 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
Distinct variable groups:   𝐴,𝑓,𝑛   𝑓,𝐹,𝑖,𝑦   𝑅,𝑓,𝑛   𝑖,𝑛,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑓,𝑖,𝑛)   𝐴(𝑥,𝑦,𝑖)   𝑅(𝑥,𝑦,𝑖)   𝐹(𝑥,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑛)   𝜓″(𝑥,𝑦,𝑓,𝑖,𝑛)

Proof of Theorem bnj126
StepHypRef Expression
1 bnj126.3 . 2 (𝜓″[𝐹 / 𝑓]𝜓′)
2 bnj126.2 . . 3 (𝜓′[1o / 𝑛]𝜓)
32sbcbii 3832 . 2 ([𝐹 / 𝑓]𝜓′[𝐹 / 𝑓][1o / 𝑛]𝜓)
4 bnj126.1 . . 3 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
5 bnj126.4 . . . 4 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
65bnj95 32023 . . 3 𝐹 ∈ V
74, 6bnj106 32027 . 2 ([𝐹 / 𝑓][1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
81, 3, 73bitri 298 1 (𝜓″ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1530  wcel 2106  wral 3142  [wsbc 3775  c0 4294  {csn 4563  cop 4569   ciun 4916  suc csuc 6190  cfv 6351  ωcom 7571  1oc1o 8089   predc-bnj14 31845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-pw 4543  df-sn 4564  df-pr 4566  df-uni 4837  df-iun 4918  df-br 5063  df-suc 6194  df-iota 6311  df-fv 6359  df-1o 8096
This theorem is referenced by:  bnj150  32035  bnj153  32039
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