Theorem mopick 2628

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		df-mo 2543	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
2		sp 2184	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))
3		pm3.45 621	. . . . . . 7 ⊢ ((𝜑 → 𝑥 = 𝑦) → ((𝜑 ∧ 𝜓) → (𝑥 = 𝑦 ∧ 𝜓)))
4	3	aleximi 1830	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
5		ax12ev2 2181	. . . . . 6 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → (𝑥 = 𝑦 → 𝜓))
6	4, 5	syl6 35	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝜑 ∧ 𝜓) → (𝑥 = 𝑦 → 𝜓)))
7	2, 6	syl5d 73	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 → 𝜓)))
8	7	exlimiv 1929	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 → 𝜓)))
9	1, 8	sylbi 217	. 2 ⊢ (∃*𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 → 𝜓)))
10	9	imp 406	1 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535  ∃wex 1777  ∃*wmo 2541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-mo 2543

This theorem is referenced by:  moexexlem  2629  eupick  2636  mopick2  2640  morex  3741  imadif  6662  metsscmetcld  25368