Theorem mo2icl 2909

Description: Theorem for inferring "at most one". (Contributed by NM, 17-Oct-1996.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem mo2icl

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfa1 1534	. . . . 5 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝐴)
2		vex 2733	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		eleq1 2233	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
4	2, 3	mpbii 147	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
5	4	imim2i 12	. . . . . 6 ⊢ ((𝜑 → 𝑥 = 𝐴) → (𝜑 → 𝐴 ∈ V))
6	5	sps 1530	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → (𝜑 → 𝐴 ∈ V))
7	1, 6	eximd 1605	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → (∃𝑥𝜑 → ∃𝑥 𝐴 ∈ V))
8		19.9v 1864	. . . 4 ⊢ (∃𝑥 𝐴 ∈ V ↔ 𝐴 ∈ V)
9	7, 8	syl6ib 160	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → (∃𝑥𝜑 → 𝐴 ∈ V))
10		eqeq2 2180	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
11	10	imbi2d 229	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴)))
12	11	albidv 1817	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝐴)))
13	12	imbi1d 230	. . . . 5 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)))
14		nfv 1521	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
15	14	mo2r 2071	. . . . . 6 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)
16	15	19.23bi 1585	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)
17	13, 16	vtoclg 2790	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑))
18	17	com12 30	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → (𝐴 ∈ V → ∃*𝑥𝜑))
19	9, 18	syld 45	. 2 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → (∃𝑥𝜑 → ∃*𝑥𝜑))
20		moabs 2068	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
21	19, 20	sylibr 133	1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1346   = wceq 1348  ∃wex 1485  ∃*wmo 2020   ∈ wcel 2141  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  invdisj  3983