Theorem mo2icl 1434

Description: Theorem for inferring "at most one".

Assertion

Ref	Expression
mo2icl	⊢ (∀x(φ → x = A) → ∃*xφ)

Distinct variable group(s):   x,A

Proof of Theorem mo2icl

Step	Hyp	Ref	Expression
1		cleq2 1110	. . . . . 6 ⊢ (y = A → (x = y ↔ x = A))
2	1	imbi2d 464	. . . . 5 ⊢ (y = A → ((φ → x = y) ↔ (φ → x = A)))
3	2	bialdv 935	. . . 4 ⊢ (y = A → (∀x(φ → x = y) ↔ ∀x(φ → x = A)))
4	3	imbi1d 465	. . 3 ⊢ (y = A → ((∀x(φ → x = y) → ∃*xφ) ↔ (∀x(φ → x = A) → ∃*xφ)))
5		19.8a 712	. . . 4 ⊢ (∀x(φ → x = y) → ∃y∀x(φ → x = y))
6		ax-17 925	. . . . 5 ⊢ (φ → ∀yφ)
7	6	mo2 1026	. . . 4 ⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))
8	5, 7	sylibr 175	. . 3 ⊢ (∀x(φ → x = y) → ∃*xφ)
9	4, 8	vtoclg 1383	. 2 ⊢ (A ∈ V → (∀x(φ → x = A) → ∃*xφ))
10		visset 1350	. . . . . . . 8 ⊢ x ∈ V
11		eleq1 1149	. . . . . . . 8 ⊢ (x = A → (x ∈ V ↔ A ∈ V))
12	10, 11	mpbii 168	. . . . . . 7 ⊢ (x = A → A ∈ V)
13	12	syl3 18	. . . . . 6 ⊢ ((φ → x = A) → (φ → A ∈ V))
14	13	con3d 87	. . . . 5 ⊢ ((φ → x = A) → (¬ A ∈ V → ¬ φ))
15	14	com12 13	. . . 4 ⊢ (¬ A ∈ V → ((φ → x = A) → ¬ φ))
16	15	19.20dv 946	. . 3 ⊢ (¬ A ∈ V → (∀x(φ → x = A) → ∀x ¬ φ))
17		alnex 716	. . . 4 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
18		exmo 1042	. . . . 5 ⊢ (∃xφ ∨ ∃*xφ)
19	18	ori 200	. . . 4 ⊢ (¬ ∃xφ → ∃*xφ)
20	17, 19	sylbi 174	. . 3 ⊢ (∀x ¬ φ → ∃*xφ)
21	16, 20	syl6 23	. 2 ⊢ (¬ A ∈ V → (∀x(φ → x = A) → ∃*xφ))
22	9, 21	pm2.61i 110	1 ⊢ (∀x(φ → x = A) → ∃*xφ)

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672  ∃wex 678   = weq 797  ∃*wmo 1008   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  aceq6b 3565

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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