Theorem bnj986 35519

Description: Technical lemma for bnj69 35574. 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
bnj986.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj986.10	⊢ 𝐷 = (ω ∖ {∅})
bnj986.15	⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))

Assertion

Ref	Expression
bnj986	⊢ (𝜒 → ∃𝑚∃𝑝𝜏)

Distinct variable group:   𝑚,𝑛,𝑝

Allowed substitution hints:   𝜑(𝑓, 𝑚, 𝑛, 𝑝)   𝜓(𝑓, 𝑚, 𝑛, 𝑝)   𝜒(𝑓, 𝑚, 𝑛, 𝑝)   𝜏(𝑓, 𝑚, 𝑛, 𝑝)   𝐷(𝑓, 𝑚, 𝑛, 𝑝)

Proof of Theorem bnj986

Step	Hyp	Ref	Expression
1		bnj986.3	. . . . . 6 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
2		bnj986.10	. . . . . . 7 ⊢ 𝐷 = (ω ∖ {∅})
3	2	bnj158 35294	. . . . . 6 ⊢ (𝑛 ∈ 𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
4	1, 3	bnj769 35327	. . . . 5 ⊢ (𝜒 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
5	4	bnj1196 35358	. . . 4 ⊢ (𝜒 → ∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚))
6		vex 3454	. . . . . 6 ⊢ 𝑛 ∈ V
7	6	sucex 7803	. . . . 5 ⊢ suc 𝑛 ∈ V
8	7	isseti 3468	. . . 4 ⊢ ∃𝑝 𝑝 = suc 𝑛
9	5, 8	jctir 530	. . 3 ⊢ (𝜒 → (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
10		exdistr 1987	. . . 4 ⊢ (∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛) ↔ ∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
11		19.41v 1982	. . . 4 ⊢ (∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛) ↔ (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
12	10, 11	bitr2i 279	. . 3 ⊢ ((∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛) ↔ ∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
13	9, 12	sylib 221	. 2 ⊢ (𝜒 → ∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
14		bnj986.15	. . . 4 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
15		df-3an 1105	. . . 4 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
16	14, 15	bitri 278	. . 3 ⊢ (𝜏 ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
17	16	2exbii 1882	. 2 ⊢ (∃𝑚∃𝑝𝜏 ↔ ∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
18	13, 17	sylibr 237	1 ⊢ (𝜒 → ∃𝑚∃𝑝𝜏)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3086   ∖ cdif 3895  ∅c0 4278  {csn 4583  suc csuc 6353   Fn wfn 6522  ωcom 7860   ∧ w-bnj17 35251

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-om 7861  df-bnj17 35252

This theorem is used by:  bnj996  35520