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Theorem bnj556 31851
Description: Technical lemma for bnj852 31872. 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
bnj556.18 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
bnj556.19 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
Assertion
Ref Expression
bnj556 (𝜂𝜎)

Proof of Theorem bnj556
StepHypRef Expression
1 vex 3420 . . . . 5 𝑝 ∈ V
21bnj216 31682 . . . 4 (𝑚 = suc 𝑝𝑝𝑚)
323anim3i 1135 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) → (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
43adantr 473 . 2 (((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω) → (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
5 bnj556.19 . . 3 (𝜂 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
6 bnj258 31658 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω))
75, 6bitri 267 . 2 (𝜂 ↔ ((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω))
8 bnj556.18 . 2 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
94, 7, 83imtr4i 284 1 (𝜂𝜎)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069   = wceq 1508  wcel 2051  suc csuc 6036  ωcom 7402  w-bnj17 31636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2752
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-v 3419  df-un 3836  df-sn 4445  df-suc 6040  df-bnj17 31637
This theorem is referenced by:  bnj557  31852  bnj561  31854  bnj562  31855
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