Theorem mopick 2023

Description: "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.)

Assertion

Ref	Expression
mopick	⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))

Proof of Theorem mopick

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1462	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑦(𝜑 ∧ 𝜓))
2		hbs1 1859	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3		hbs1 1859	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜓 → ∀𝑥[𝑦 / 𝑥]𝜓)
4	2, 3	hban 1482	. . . 4 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → ∀𝑥([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
5		sbequ12 1698	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
6		sbequ12 1698	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑥]𝜓))
7	5, 6	anbi12d 457	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓)))
8	1, 4, 7	cbvexh 1682	. . 3 ⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑦([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
9		ax-17 1462	. . . . . . 7 ⊢ (𝜑 → ∀𝑦𝜑)
10	9	mo3h 1998	. . . . . 6 ⊢ (∃*𝑥𝜑 ↔ ∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
11		ax-4 1443	. . . . . . 7 ⊢ (∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
12	11	sps 1473	. . . . . 6 ⊢ (∀𝑥∀𝑦((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
13	10, 12	sylbi 119	. . . . 5 ⊢ (∃*𝑥𝜑 → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦))
14		sbequ2 1696	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜓 → 𝜓))
15	14	imim2i 12	. . . . . . . 8 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → ([𝑦 / 𝑥]𝜓 → 𝜓)))
16	15	expd 254	. . . . . . 7 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑 → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → 𝜓))))
17	16	com4t 84	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑 → 𝜓))))
18	17	imp 122	. . . . 5 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) → (𝜑 → 𝜓)))
19	13, 18	syl5 32	. . . 4 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → (∃*𝑥𝜑 → (𝜑 → 𝜓)))
20	19	exlimiv 1532	. . 3 ⊢ (∃𝑦([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → (∃*𝑥𝜑 → (𝜑 → 𝜓)))
21	8, 20	sylbi 119	. 2 ⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃*𝑥𝜑 → (𝜑 → 𝜓)))
22	21	impcom 123	1 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1285  ∃wex 1424  [wsb 1689  ∃*wmo 1946

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471

This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949

This theorem is referenced by:  eupick  2024  mopick2  2028  moexexdc  2029  euexex  2030  morex  2790  imadif  5059  funimaexglem  5062