Theorem mo2icl 3644

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 2750	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
2	1	imbi2d 340	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴)))
3	2	albidv 1924	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝐴)))
4	3	imbi1d 341	. . 3 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)))
5		equequ2 2030	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
6	5	imbi2d 340	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝑧)))
7	6	albidv 1924	. . . . 5 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑧)))
8	7	19.8aw 2054	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
9		df-mo 2540	. . . 4 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
10	8, 9	sylibr 233	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)
11	4, 10	vtoclg 3495	. 2 ⊢ (𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑))
12		eqvisset 3439	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
13	12	imim2i 16	. . . . 5 ⊢ ((𝜑 → 𝑥 = 𝐴) → (𝜑 → 𝐴 ∈ V))
14	13	con3rr3 155	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝜑 → 𝑥 = 𝐴) → ¬ 𝜑))
15	14	alimdv 1920	. . 3 ⊢ (¬ 𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∀𝑥 ¬ 𝜑))
16		alnex 1785	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
17		nexmo 2541	. . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
18	16, 17	sylbi 216	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∃*𝑥𝜑)
19	15, 18	syl6 35	. 2 ⊢ (¬ 𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑))
20	11, 19	pm2.61i 182	1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃*wmo 2538  Vcvv 3422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424

This theorem is referenced by:  invdisj  5054  reusv1  5315  reusv2lem1  5316  opabiotafun  6831  fseqenlem2  9712  dfac2b  9817  imasaddfnlem  17156  imasvscafn  17165  bnj149  32755