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Theorem bnj556 33911
Description: Technical lemma for bnj852 33932. 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 3479 . . . . 5 𝑝 ∈ V
21bnj216 33743 . . . 4 (𝑚 = suc 𝑝𝑝𝑚)
323anim3i 1155 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) → (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
43adantr 482 . 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 33719 . . 3 ((𝑚𝐷𝑛 = suc 𝑚𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω))
75, 6bitri 275 . 2 (𝜂 ↔ ((𝑚𝐷𝑛 = suc 𝑚𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω))
8 bnj556.18 . 2 (𝜎 ↔ (𝑚𝐷𝑛 = suc 𝑚𝑝𝑚))
94, 7, 83imtr4i 292 1 (𝜂𝜎)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  suc csuc 6367  ωcom 7855  w-bnj17 33697
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-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-sn 4630  df-suc 6371  df-bnj17 33698
This theorem is referenced by:  bnj557  33912  bnj561  33914  bnj562  33915
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