Theorem mo2icl 2985

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 1589	. . . . 5 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝐴)
2		vex 2805	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		eleq1 2294	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V))
4	2, 3	mpbii 148	. . . . . . 7 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
5	4	imim2i 12	. . . . . 6 ⊢ ((𝜑 → 𝑥 = 𝐴) → (𝜑 → 𝐴 ∈ V))
6	5	sps 1585	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → (𝜑 → 𝐴 ∈ V))
7	1, 6	eximd 1660	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → (∃𝑥𝜑 → ∃𝑥 𝐴 ∈ V))
8		19.9v 1919	. . . 4 ⊢ (∃𝑥 𝐴 ∈ V ↔ 𝐴 ∈ V)
9	7, 8	imbitrdi 161	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → (∃𝑥𝜑 → 𝐴 ∈ V))
10		eqeq2 2241	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
11	10	imbi2d 230	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴)))
12	11	albidv 1872	. . . . . 6 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝐴)))
13	12	imbi1d 231	. . . . 5 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)))
14		nfv 1576	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
15	14	mo2r 2132	. . . . . 6 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)
16	15	19.23bi 1640	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)
17	13, 16	vtoclg 2864	. . . 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 2129	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
21	19, 20	sylibr 134	1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1395   = wceq 1397  ∃wex 1540  ∃*wmo 2080   ∈ wcel 2202  Vcvv 2802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804

This theorem is referenced by:  invdisj  4081  imasaddfnlemg  13399