Theorem bnj556 30970

Description: Technical lemma for bnj852 30991. 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 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
bnj556.19	⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))

Assertion

Ref	Expression
bnj556	⊢ (𝜂 → 𝜎)

Proof of Theorem bnj556

Step	Hyp	Ref	Expression
1		vex 3203	. . . . 5 ⊢ 𝑝 ∈ V
2	1	bnj216 30800	. . . 4 ⊢ (𝑚 = suc 𝑝 → 𝑝 ∈ 𝑚)
3	2	3anim3i 1250	. . 3 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝) → (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
4	3	adantr 481	. 2 ⊢ (((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω) → (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
5		bnj556.19	. . 3 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
6		bnj258 30774	. . 3 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω))
7	5, 6	bitri 264	. 2 ⊢ (𝜂 ↔ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑚 = suc 𝑝) ∧ 𝑝 ∈ ω))
8		bnj556.18	. 2 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
9	4, 7, 8	3imtr4i 281	1 ⊢ (𝜂 → 𝜎)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  suc csuc 5725  ωcom 7065   ∧ w-bnj17 30752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-suc 5729  df-bnj17 30753

This theorem is referenced by:  bnj557  30971  bnj561  30973  bnj562  30974