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Theorem bnj556 32232
 Description: Technical lemma for bnj852 32253. 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 3483 . . . . 5 𝑝 ∈ V
21bnj216 32062 . . . 4 (𝑚 = suc 𝑝𝑝𝑚)
323anim3i 1151 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) → (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
43adantr 484 . 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 32038 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω))
75, 6bitri 278 . 2 (𝜂 ↔ ((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω))
8 bnj556.18 . 2 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
94, 7, 83imtr4i 295 1 (𝜂𝜎)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  suc csuc 6180  ωcom 7574   ∧ w-bnj17 32016 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924  df-sn 4551  df-suc 6184  df-bnj17 32017 This theorem is referenced by:  bnj557  32233  bnj561  32235  bnj562  32236
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