Theorem mo2icl 1919

Description: Theorem for inferring "at most one."

Assertion

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

Distinct variable group:   x,A

Proof of Theorem mo2icl

Step	Hyp	Ref	Expression
1		eqeq2 1481	. . . . . 6 ⊢ (y = A → (x = y ↔ x = A))
2	1	imbi2d 611	. . . . 5 ⊢ (y = A → ((φ → x = y) ↔ (φ → x = A)))
3	2	albidv 1276	. . . 4 ⊢ (y = A → (∀x(φ → x = y) ↔ ∀x(φ → x = A)))
4	3	imbi1d 612	. . 3 ⊢ (y = A → ((∀x(φ → x = y) → ∃*xφ) ↔ (∀x(φ → x = A) → ∃*xφ)))
5		19.8a 1027	. . . 4 ⊢ (∀x(φ → x = y) → ∃y∀x(φ → x = y))
6		ax-17 969	. . . . 5 ⊢ (φ → ∀yφ)
7	6	mo2 1398	. . . 4 ⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))
8	5, 7	sylibr 200	. . 3 ⊢ (∀x(φ → x = y) → ∃*xφ)
9	4, 8	vtoclg 1843	. 2 ⊢ (A ∈ V → (∀x(φ → x = A) → ∃*xφ))
10		visset 1809	. . . . . . . 8 ⊢ x ∈ V
11		eleq1 1531	. . . . . . . 8 ⊢ (x = A → (x ∈ V ↔ A ∈ V))
12	10, 11	mpbii 193	. . . . . . 7 ⊢ (x = A → A ∈ V)
13	12	imim2i 17	. . . . . 6 ⊢ ((φ → x = A) → (φ → A ∈ V))
14	13	con3d 95	. . . . 5 ⊢ ((φ → x = A) → (¬ A ∈ V → ¬ φ))
15	14	com12 11	. . . 4 ⊢ (¬ A ∈ V → ((φ → x = A) → ¬ φ))
16	15	19.20dv 1287	. . 3 ⊢ (¬ A ∈ V → (∀x(φ → x = A) → ∀x ¬ φ))
17		alnex 1031	. . . 4 ⊢ (∀x ¬ φ ↔ ¬ ∃xφ)
18		exmo 1414	. . . . 5 ⊢ (∃xφ ⋁ ∃*xφ)
19	18	ori 230	. . . 4 ⊢ (¬ ∃xφ → ∃*xφ)
20	17, 19	sylbi 199	. . 3 ⊢ (∀x ¬ φ → ∃*xφ)
21	16, 20	syl6 22	. 2 ⊢ (¬ A ∈ V → (∀x(φ → x = A) → ∃*xφ))
22	9, 21	pm2.61i 126	1 ⊢ (∀x(φ → x = A) → ∃*xφ)

Syntax hints:  ¬ wn 2   → wi 3  ∀wal 952   = wceq 954   ∈ wcel 956  ∃wex 978  ∃*wmo 1379  Vcvv 1807

This theorem is referenced by:  aceq6b 4722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808

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