Theorem mosubopt 4612

Description: "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-Aug-2007.)

Assertion

Ref	Expression
mosubopt	⊢ (∀y∀z∃*xφ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))

Distinct variable group:   x,y,z,A

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem mosubopt

Step	Hyp	Ref	Expression
1		nfa1 1788	. . 3 ⊢ Ⅎy∀y∀z∃*xφ
2		nfe1 1732	. . . 4 ⊢ Ⅎy∃y∃z(A = ⟨y, z⟩ ∧ φ)
3	2	nfmo 2221	. . 3 ⊢ Ⅎy∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)
4		nfa1 1788	. . . . 5 ⊢ Ⅎz∀z∃*xφ
5		nfe1 1732	. . . . . . 7 ⊢ Ⅎz∃z(A = ⟨y, z⟩ ∧ φ)
6	5	nfex 1843	. . . . . 6 ⊢ Ⅎz∃y∃z(A = ⟨y, z⟩ ∧ φ)
7	6	nfmo 2221	. . . . 5 ⊢ Ⅎz∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)
8		copsexg 4607	. . . . . . . 8 ⊢ (A = ⟨y, z⟩ → (φ ↔ ∃y∃z(A = ⟨y, z⟩ ∧ φ)))
9	8	mobidv 2239	. . . . . . 7 ⊢ (A = ⟨y, z⟩ → (∃*xφ ↔ ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
10	9	biimpcd 215	. . . . . 6 ⊢ (∃*xφ → (A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
11	10	sps 1754	. . . . 5 ⊢ (∀z∃*xφ → (A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
12	4, 7, 11	exlimd 1806	. . . 4 ⊢ (∀z∃*xφ → (∃z A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
13	12	sps 1754	. . 3 ⊢ (∀y∀z∃*xφ → (∃z A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
14	1, 3, 13	exlimd 1806	. 2 ⊢ (∀y∀z∃*xφ → (∃y∃z A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ)))
15		simpl 443	. . . . . 6 ⊢ ((A = ⟨y, z⟩ ∧ φ) → A = ⟨y, z⟩)
16	15	2eximi 1577	. . . . 5 ⊢ (∃y∃z(A = ⟨y, z⟩ ∧ φ) → ∃y∃z A = ⟨y, z⟩)
17	16	exlimiv 1634	. . . 4 ⊢ (∃x∃y∃z(A = ⟨y, z⟩ ∧ φ) → ∃y∃z A = ⟨y, z⟩)
18	17	con3i 127	. . 3 ⊢ (¬ ∃y∃z A = ⟨y, z⟩ → ¬ ∃x∃y∃z(A = ⟨y, z⟩ ∧ φ))
19		exmo 2249	. . . 4 ⊢ (∃x∃y∃z(A = ⟨y, z⟩ ∧ φ) ∨ ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))
20	19	ori 364	. . 3 ⊢ (¬ ∃x∃y∃z(A = ⟨y, z⟩ ∧ φ) → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))
21	18, 20	syl 15	. 2 ⊢ (¬ ∃y∃z A = ⟨y, z⟩ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))
22	14, 21	pm2.61d1 151	1 ⊢ (∀y∀z∃*xφ → ∃*x∃y∃z(A = ⟨y, z⟩ ∧ φ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃*wmo 2205  ⟨cop 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568

This theorem is referenced by:  mosubop  4613  funoprabg  5583