Theorem mo2icl 2916

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 1541	. . . . 5 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝑥 = 𝐴)
2		vex 2740	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		eleq1 2240	. . . . . . . 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 1537	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → (𝜑 → 𝐴 ∈ V))
7	1, 6	eximd 1612	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → (∃𝑥𝜑 → ∃𝑥 𝐴 ∈ V))
8		19.9v 1871	. . . 4 ⊢ (∃𝑥 𝐴 ∈ V ↔ 𝐴 ∈ V)
9	7, 8	imbitrdi 161	. . 3 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → (∃𝑥𝜑 → 𝐴 ∈ V))
10		eqeq2 2187	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (𝑥 = 𝑦 ↔ 𝑥 = 𝐴))
11	10	imbi2d 230	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝐴)))
12	11	albidv 1824	. . . . . 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 1528	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
15	14	mo2r 2078	. . . . . 6 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)
16	15	19.23bi 1592	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → ∃*𝑥𝜑)
17	13, 16	vtoclg 2797	. . . 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 2075	. 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃*𝑥𝜑))
21	19, 20	sylibr 134	1 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝐴) → ∃*𝑥𝜑)

Syntax hints:   → wi 4  ∀wal 1351   = wceq 1353  ∃wex 1492  ∃*wmo 2027   ∈ wcel 2148  Vcvv 2737

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739

This theorem is referenced by:  invdisj  3997  imasaddfnlemg  12734