Theorem mo2icl 3680

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 variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq2 2777	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
2	1	imbi2d 343	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴)))
3	2	albidv 1943	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝐴)))
4	3	imbi1d 344	. . 3 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)))
5		equequ2 2049	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
6	5	imbi2d 343	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝑧)))
7	6	albidv 1943	. . . . 5 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑧)))
8	7	19.8aw 2075	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
9		dfmo 2570	. . . 4 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
10	8, 9	sylibr 237	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)
11	4, 10	vtoclg 3525	. 2 ⊢ (𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑))
12		eqvisset 3477	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
13	12	imim2i 17	. . . . 5 ⊢ ((𝜑 → 𝑥 = 𝐴) → (𝜑 → 𝐴 ∈ V))
14	13	con3rr3 156	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝜑 → 𝑥 = 𝐴) → ¬ 𝜑))
15	14	alimdv 1939	. . 3 ⊢ (¬ 𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∀𝑥 ¬ 𝜑))
16		alnex 1804	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
17		nexmo 2571	. . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
18	16, 17	sylbi 220	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∃*𝑥𝜑)
19	15, 18	syl6 36	. 2 ⊢ (¬ 𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑))
20	11, 19	pm2.61i 184	1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1561   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∃*wmo 2567  Vcvv 3457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459

This theorem is referenced by:  invdisj  5091  reusv1  5359  reusv2lem1  5360  opabiotafun  6951  fseqenlem2  9997  dfac2b  10102  imasaddfnlem  17572  imasvscafn  17581  bnj149  35180