Theorem mopick 1054

Description: "At most one" picks a variable value, eliminating an existential quantifier.

Assertion

Ref	Expression
mopick	⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))

Proof of Theorem mopick

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ ((φ ∧ ψ) → ∀y(φ ∧ ψ))
2		hbs1 986	. . . . . 6 ⊢ ([y / x]φ → ∀x[y / x]φ)
3		hbs1 986	. . . . . 6 ⊢ ([y / x]ψ → ∀x[y / x]ψ)
4	2, 3	hban 704	. . . . 5 ⊢ (([y / x]φ ∧ [y / x]ψ) → ∀x([y / x]φ ∧ [y / x]ψ))
5		sbequ12 865	. . . . . 6 ⊢ (x = y → (φ ↔ [y / x]φ))
6		sbequ12 865	. . . . . 6 ⊢ (x = y → (ψ ↔ [y / x]ψ))
7	5, 6	anbi12d 476	. . . . 5 ⊢ (x = y → ((φ ∧ ψ) ↔ ([y / x]φ ∧ [y / x]ψ)))
8	1, 4, 7	cbvex 849	. . . 4 ⊢ (∃x(φ ∧ ψ) ↔ ∃y([y / x]φ ∧ [y / x]ψ))
9		sbequ2 864	. . . . . . . . . 10 ⊢ (x = y → ([y / x]ψ → ψ))
10	9	syl3 18	. . . . . . . . 9 ⊢ (((φ ∧ [y / x]φ) → x = y) → ((φ ∧ [y / x]φ) → ([y / x]ψ → ψ)))
11	10	exp3a 292	. . . . . . . 8 ⊢ (((φ ∧ [y / x]φ) → x = y) → (φ → ([y / x]φ → ([y / x]ψ → ψ))))
12	11	com4t 40	. . . . . . 7 ⊢ ([y / x]φ → ([y / x]ψ → (((φ ∧ [y / x]φ) → x = y) → (φ → ψ))))
13	12	imp 277	. . . . . 6 ⊢ (([y / x]φ ∧ [y / x]ψ) → (((φ ∧ [y / x]φ) → x = y) → (φ → ψ)))
14		ax-17 925	. . . . . . . 8 ⊢ (φ → ∀yφ)
15	14	mo3 1027	. . . . . . 7 ⊢ (∃*xφ ↔ ∀x∀y((φ ∧ [y / x]φ) → x = y))
16		ax-4 673	. . . . . . . 8 ⊢ (∀y((φ ∧ [y / x]φ) → x = y) → ((φ ∧ [y / x]φ) → x = y))
17	16	a4s 682	. . . . . . 7 ⊢ (∀x∀y((φ ∧ [y / x]φ) → x = y) → ((φ ∧ [y / x]φ) → x = y))
18	15, 17	sylbi 174	. . . . . 6 ⊢ (∃*xφ → ((φ ∧ [y / x]φ) → x = y))
19	13, 18	syl5 22	. . . . 5 ⊢ (([y / x]φ ∧ [y / x]ψ) → (∃*xφ → (φ → ψ)))
20	19	19.23aiv 952	. . . 4 ⊢ (∃y([y / x]φ ∧ [y / x]ψ) → (∃*xφ → (φ → ψ)))
21	8, 20	sylbi 174	. . 3 ⊢ (∃x(φ ∧ ψ) → (∃*xφ → (φ → ψ)))
22	21	com12 13	. 2 ⊢ (∃*xφ → (∃x(φ ∧ ψ) → (φ → ψ)))
23	22	imp 277	1 ⊢ ((∃*xφ ∧ ∃x(φ ∧ ψ)) → (φ → ψ))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  [wsb 852  ∃*wmo 1008

This theorem is referenced by:  eupick 1055  mopick2 1057  moexex 1058  imadif 2714

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010

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