Theorem mopick 1975

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 1416	. . . 4 ⊢ ((φ ∧ ψ) → ∀y(φ ∧ ψ))
2		hbs1 1811	. . . . 5 ⊢ ([y / x]φ → ∀x[y / x]φ)
3		hbs1 1811	. . . . 5 ⊢ ([y / x]ψ → ∀x[y / x]ψ)
4	2, 3	hban 1436	. . . 4 ⊢ (([y / x]φ ∧ [y / x]ψ) → ∀x([y / x]φ ∧ [y / x]ψ))
5		sbequ12 1651	. . . . 5 ⊢ (x = y → (φ ↔ [y / x]φ))
6		sbequ12 1651	. . . . 5 ⊢ (x = y → (ψ ↔ [y / x]ψ))
7	5, 6	anbi12d 442	. . . 4 ⊢ (x = y → ((φ ∧ ψ) ↔ ([y / x]φ ∧ [y / x]ψ)))
8	1, 4, 7	cbvexh 1635	. . 3 ⊢ (∃x(φ ∧ ψ) ↔ ∃y([y / x]φ ∧ [y / x]ψ))
9		ax-17 1416	. . . . . . 7 ⊢ (φ → ∀yφ)
10	9	mo3h 1950	. . . . . 6 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))
11		ax-4 1397	. . . . . . 7 ⊢ (∀y((φ ∧ [y / x]φ) → x = y) → ((φ ∧ [y / x]φ) → x = y))
12	11	sps 1427	. . . . . 6 ⊢ (∀x∀y((φ ∧ [y / x]φ) → x = y) → ((φ ∧ [y / x]φ) → x = y))
13	10, 12	sylbi 114	. . . . 5 ⊢ (∃*xφ → ((φ ∧ [y / x]φ) → x = y))
14		sbequ2 1649	. . . . . . . . 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 245	. . . . . . 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 115	. . . . 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 1486	. . 3 ⊢ (∃y([y / x]φ ∧ [y / x]ψ) → (∃*xφ → (φ → ψ)))
21	8, 20	sylbi 114	. 2 ⊢ (∃x(φ ∧ ψ) → (∃*xφ → (φ → ψ)))
22	21	impcom 116	1 ⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240  ∃wex 1378  [wsb 1642  ∃*wmo 1898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901

This theorem is referenced by:  eupick  1976  mopick2  1980  moexexdc  1981  euexex  1982  morex  2719  imadif  4922  funimaexglem  4925