Theorem mopick 2523

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		mo2v 2465	. . 3 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
2		sp 2041	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦))
3		pm3.45 875	. . . . . . 7 ⊢ ((𝜑 → 𝑥 = 𝑦) → ((𝜑 ∧ 𝜓) → (𝑥 = 𝑦 ∧ 𝜓)))
4	3	aleximi 1749	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)))
5		sb56 2136	. . . . . . 7 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜓))
6		sp 2041	. . . . . . 7 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜓) → (𝑥 = 𝑦 → 𝜓))
7	5, 6	sylbi 206	. . . . . 6 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) → (𝑥 = 𝑦 → 𝜓))
8	4, 7	syl6 34	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝜑 ∧ 𝜓) → (𝑥 = 𝑦 → 𝜓)))
9	2, 8	syl5d 71	. . . 4 ⊢ (∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 → 𝜓)))
10	9	exlimiv 1845	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 → 𝜓)))
11	1, 10	sylbi 206	. 2 ⊢ (∃*𝑥𝜑 → (∃𝑥(𝜑 ∧ 𝜓) → (𝜑 → 𝜓)))
12	11	imp 444	1 ⊢ ((∃*𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → (𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695  ∃*wmo 2459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463

This theorem is referenced by:  eupick  2524  mopick2  2528  moexex  2529  morex  3357  imadif  5887  cmetss  22921