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Theorem bnj126 33915
Description: Technical lemma for bnj150 33918. 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 3838 . 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 33906 . . 3 𝐹 ∈ V
74, 6bnj106 33910 . 2 ([𝐹 / 𝑓][1o / 𝑛]𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 1o → (𝐹‘suc 𝑖) = 𝑦 ∈ (𝐹𝑖) pred(𝑦, 𝐴, 𝑅)))
81, 3, 73bitri 297 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 205   = wceq 1542  wcel 2107  wral 3062  [wsbc 3778  c0 4323  {csn 4629  cop 4635   ciun 4998  suc csuc 6367  cfv 6544  ωcom 7855  1oc1o 8459   predc-bnj14 33730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-pw 4605  df-sn 4630  df-pr 4632  df-uni 4910  df-iun 5000  df-br 5150  df-suc 6371  df-iota 6496  df-fv 6552  df-1o 8466
This theorem is referenced by:  bnj150  33918  bnj153  33922
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