Theorem mopick 2650

Description: "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.)

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		dfmo 2565	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
2		sp 2219	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))
3		pm3.45 634	. . . . . . 7 ⊢ ((𝜑 → 𝑥 = 𝑦) → ((𝜑 ∧ 𝜓) → (𝑥 = 𝑦 ∧ 𝜓)))
4	3	aleximi 1864	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
5		ax12ev2 2215	. . . . . 6 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → (𝑥 = 𝑦 → 𝜓))
6	4, 5	syl6 36	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝜑 ∧ 𝜓) → (𝑥 = 𝑦 → 𝜓)))
7	2, 6	syl5d 74	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 → 𝜓)))
8	7	exlimiv 1962	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 → 𝜓)))
9	1, 8	sylbi 220	. 2 ⊢ (∃*𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 → 𝜓)))
10	9	imp 412	1 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567  ∃wex 1811  ∃*wmo 2562

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-mo 2564

This theorem is used by:  moexexlem  2651  eupick  2658  mopick2  2662  morex  3676  imadif  6624  metsscmetcld  25641