Theorem mo2icl 3367

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		eqeq2 2632	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
2	1	imbi2d 330	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴)))
3	2	albidv 1846	. . . 4 ⊢ (𝑦 = 𝐴 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝐴)))
4	3	imbi1d 331	. . 3 ⊢ (𝑦 = 𝐴 → ((∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑) ↔ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)))
5		19.8a 2049	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
6		mo2v 2476	. . . 4 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
7	5, 6	sylibr 224	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)
8	4, 7	vtoclg 3252	. 2 ⊢ (𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑))
9		eqvisset 3197	. . . . . 6 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V)
10	9	imim2i 16	. . . . 5 ⊢ ((𝜑 → 𝑥 = 𝐴) → (𝜑 → 𝐴 ∈ V))
11	10	con3rr3 151	. . . 4 ⊢ (¬ 𝐴 ∈ V → ((𝜑 → 𝑥 = 𝐴) → ¬ 𝜑))
12	11	alimdv 1842	. . 3 ⊢ (¬ 𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∀𝑥 ¬ 𝜑))
13		alnex 1703	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
14		exmo 2494	. . . . 5 ⊢ (∃𝑥𝜑 ∨ ∃*𝑥𝜑)
15	14	ori 390	. . . 4 ⊢ (¬ ∃𝑥𝜑 → ∃*𝑥𝜑)
16	13, 15	sylbi 207	. . 3 ⊢ (∀𝑥 ¬ 𝜑 → ∃*𝑥𝜑)
17	12, 16	syl6 35	. 2 ⊢ (¬ 𝐴 ∈ V → (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑))
18	8, 17	pm2.61i 176	1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1478   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∃*wmo 2470  Vcvv 3186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-12 2044  ax-ext 2601

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3188

This theorem is referenced by:  invdisj  4601  reusv1  4826  reusv2lem1  4828  opabiotafun  6216  fseqenlem2  8792  dfac2  8897  imasaddfnlem  16109  imasvscafn  16118  bnj149  30650