Theorem bnj986 34586

Description: Technical lemma for bnj69 34641. 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 34360	. . . . . 6 ⊢ (𝑛 ∈ 𝐷 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
4	1, 3	bnj769 34393	. . . . 5 ⊢ (𝜒 → ∃𝑚 ∈ ω 𝑛 = suc 𝑚)
5	4	bnj1196 34425	. . . 4 ⊢ (𝜒 → ∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚))
6		vex 3475	. . . . . 6 ⊢ 𝑛 ∈ V
7	6	sucex 7809	. . . . 5 ⊢ suc 𝑛 ∈ V
8	7	isseti 3487	. . . 4 ⊢ ∃𝑝 𝑝 = suc 𝑛
9	5, 8	jctir 520	. . 3 ⊢ (𝜒 → (∃𝑚(𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
10		exdistr 1951	. . . 4 ⊢ (∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛) ↔ ∃𝑚((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ ∃𝑝 𝑝 = suc 𝑛))
11		19.41v 1946	. . . 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 276	. . 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 217	. 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 1087	. . . 4 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
16	14, 15	bitri 275	. . 3 ⊢ (𝜏 ↔ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
17	16	2exbii 1844	. 2 ⊢ (∃𝑚∃𝑝𝜏 ↔ ∃𝑚∃𝑝((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚) ∧ 𝑝 = suc 𝑛))
18	13, 17	sylibr 233	1 ⊢ (𝜒 → ∃𝑚∃𝑝𝜏)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∃wrex 3067   ∖ cdif 3944  ∅c0 4323  {csn 4629  suc csuc 6371   Fn wfn 6543  ωcom 7870   ∧ w-bnj17 34317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-om 7871  df-bnj17 34318

This theorem is referenced by:  bnj996  34587