Theorem mopick 2266

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		nfv 1619	. . . 4 ⊢ Ⅎy(φ ∧ ψ)
2		nfs1v 2106	. . . . 5 ⊢ Ⅎx[y / x]φ
3		nfs1v 2106	. . . . 5 ⊢ Ⅎx[y / x]ψ
4	2, 3	nfan 1824	. . . 4 ⊢ Ⅎx([y / x]φ ∧ [y / x]ψ)
5		sbequ12 1919	. . . . 5 ⊢ (x = y → (φ ↔ [y / x]φ))
6		sbequ12 1919	. . . . 5 ⊢ (x = y → (ψ ↔ [y / x]ψ))
7	5, 6	anbi12d 691	. . . 4 ⊢ (x = y → ((φ ∧ ψ) ↔ ([y / x]φ ∧ [y / x]ψ)))
8	1, 4, 7	cbvex 1985	. . 3 ⊢ (∃x(φ ∧ ψ) ↔ ∃y([y / x]φ ∧ [y / x]ψ))
9		nfv 1619	. . . . . . 7 ⊢ Ⅎyφ
10	9	mo3 2235	. . . . . 6 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))
11		sp 1747	. . . . . . 7 ⊢ (∀y((φ ∧ [y / x]φ) → x = y) → ((φ ∧ [y / x]φ) → x = y))
12	11	sps 1754	. . . . . 6 ⊢ (∀x∀y((φ ∧ [y / x]φ) → x = y) → ((φ ∧ [y / x]φ) → x = y))
13	10, 12	sylbi 187	. . . . 5 ⊢ (∃*xφ → ((φ ∧ [y / x]φ) → x = y))
14		sbequ2 1650	. . . . . . . . 9 ⊢ (x = y → ([y / x]ψ → ψ))
15	14	imim2i 13	. . . . . . . 8 ⊢ (((φ ∧ [y / x]φ) → x = y) → ((φ ∧ [y / x]φ) → ([y / x]ψ → ψ)))
16	15	exp3a 425	. . . . . . 7 ⊢ (((φ ∧ [y / x]φ) → x = y) → (φ → ([y / x]φ → ([y / x]ψ → ψ))))
17	16	com4t 79	. . . . . 6 ⊢ ([y / x]φ → ([y / x]ψ → (((φ ∧ [y / x]φ) → x = y) → (φ → ψ))))
18	17	imp 418	. . . . 5 ⊢ (([y / x]φ ∧ [y / x]ψ) → (((φ ∧ [y / x]φ) → x = y) → (φ → ψ)))
19	13, 18	syl5 28	. . . 4 ⊢ (([y / x]φ ∧ [y / x]ψ) → (∃*xφ → (φ → ψ)))
20	19	exlimiv 1634	. . 3 ⊢ (∃y([y / x]φ ∧ [y / x]ψ) → (∃*xφ → (φ → ψ)))
21	8, 20	sylbi 187	. 2 ⊢ (∃x(φ ∧ ψ) → (∃*xφ → (φ → ψ)))
22	21	impcom 419	1 ⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642  [wsb 1648  ∃*wmo 2205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by:  eupick  2267  mopick2  2271  moexex  2273  morex  3021  imadif  5172